P. 26 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2501-2600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	moabs 2501	Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
Theorem	exmoeu 2502	Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)
⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))
Theorem	sb8eu 2503	Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb8mo 2504	Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
Theorem	cbveu 2505	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Theorem	cbvmo 2506	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Theorem	mo3 2507*	Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	mo 2508*	Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	eu2 2509*	An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Theorem	eu1 2510*	An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
Theorem	euexALT 2511	Alternate proof of euex 2494. Shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
Theorem	euor 2512	Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))
Theorem	euorv 2513*	Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))
Theorem	euor2 2514	Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))
Theorem	sbmo 2515*	Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Theorem	mo4f 2516*	"At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	mo4 2517*	"At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	eu4 2518*	Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
Theorem	moim 2519	"At most one" reverses implication. (Contributed by NM, 22-Apr-1995.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
Theorem	moimi 2520	"At most one" reverses implication. (Contributed by NM, 15-Feb-2006.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)
Theorem	moa1 2521	If an implication holds for at most one value, then its consequent holds for at most one value. See also ala1 1741 and exa1 1765. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (Revised by BJ, 29-Mar-2021.)
⊢ (∃*𝑥(𝜑 → 𝜓) → ∃*𝑥𝜓)
Theorem	euimmo 2522	Uniqueness implies "at most one" through reverse implication. (Contributed by NM, 22-Apr-1995.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
Theorem	euim 2523	Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
Theorem	moan 2524	"At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
Theorem	moani 2525	"At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
⊢ ∃*𝑥𝜑    ⇒   ⊢ ∃*𝑥(𝜓 ∧ 𝜑)
Theorem	moor 2526	"At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑)
Theorem	mooran1 2527	"At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
Theorem	mooran2 2528	"At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))
Theorem	moanim 2529	Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	euan 2530	Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Theorem	moanimv 2531*	Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)
⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	moanmo 2532	Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
Theorem	moaneu 2533	Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Theorem	euanv 2534*	Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Theorem	mopick 2535	"At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
Theorem	eupick 2536	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing 𝑥 such that 𝜑 is true, and there is also an 𝑥 (actually the same one) such that 𝜑 and 𝜓 are both true, then 𝜑 implies 𝜓 regardless of 𝑥. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
Theorem	eupicka 2537	Version of eupick 2536 with closed formulas. (Contributed by NM, 6-Sep-2008.)
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))
Theorem	eupickb 2538	Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))
Theorem	eupickbi 2539	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
Theorem	mopick2 2540	"At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1797. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	moexex 2541	"At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
Theorem	moexexv 2542*	"At most one" double quantification. (Contributed by NM, 26-Jan-1997.)
⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
Theorem	2moex 2543	Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Theorem	2euex 2544	Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Theorem	2eumo 2545	Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
Theorem	2eu2ex 2546	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
Theorem	2moswap 2547	A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))
Theorem	2euswap 2548	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))
Theorem	2exeu 2549	Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Theorem	2mo2 2550*	This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair 𝑥 and 𝑦". Note: this is not expressed by ∃*𝑥∃*𝑦𝜑). 2eu4 2556 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)
⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2mo 2551*	Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)
⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2mos 2552*	Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2eu1 2553	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
Theorem	2eu2 2554	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑))
Theorem	2eu3 2555	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
⊢ (∀𝑥∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
Theorem	2eu4 2556*	This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2eu1 2553 for a condition under which the naive definition holds and 2exeu 2549 for a one-way implication. See 2eu5 2557 and 2eu8 2560 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	2eu5 2557*	An alternate definition of double existential uniqueness (see 2eu4 2556). A mistake sometimes made in the literature is to use ∃!𝑥∃!𝑦 to mean "exactly one 𝑥 and exactly one 𝑦." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining ∀𝑥∃*𝑦𝜑 as an additional condition. The correct definition apparently has never been published. (∃* means "exists at most one."). (Contributed by NM, 26-Oct-2003.)
⊢ ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	2eu6 2558*	Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2eu7 2559	Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
Theorem	2eu8 2560	Two equivalent expressions for double existential uniqueness. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥∃!𝑦 using 2eu7 2559. (Contributed by NM, 20-Feb-2005.)
⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))
Theorem	exists1 2561*	Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4857. (Contributed by NM, 5-Apr-2004.)
⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	exists2 2562	A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
1.7  Other axiomatizations related to classical predicate calculus
1.7.1  Aristotelian logic: Assertic syllogisms Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems. In antiquity Aristotelian logic and Stoic logic (see mptnan 1693) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe. This section models this system (including later refinements). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order. "There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation. We instead translate each Aristotelian syllogism into an inference rule, and each rule is defined using standard predicate logic notation and predicates. The predicates are represented by wff variables that may depend on the quantified variable 𝑥. Our translation is essentially identical to the one used in Rini page 18, Table 2 "Non-Modal Syllogisms in Lower Predicate Calculus (LPC)", which uses standard predicate logic with predicates. Rini states, "the crucial point is that we capture the meaning Aristotle intends, and the method by which we represent that meaning is less important." There are two differences: we make the existence criteria explicit, and we use 𝜑, 𝜓, and 𝜒 in the order they appear (a common Metamath convention). Patzig also uses standard predicate logic notation and predicates (though he interprets them as conditional propositions, not as inference rules); see Gunther Patzig, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the approach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/quantification/. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no 𝜑 is 𝜓 " are consistently translated as ∀𝑥(𝜑 → ¬ 𝜓). These can also be expressed as ¬ ∃𝑥(𝜑 ∧ 𝜓), per alinexa 1770. We translate "all 𝜑 is 𝜓 " to ∀𝑥(𝜑 → 𝜓), "some 𝜑 is 𝜓 " to ∃𝑥(𝜑 ∧ 𝜓), and "some 𝜑 is not 𝜓 " to ∃𝑥(𝜑 ∧ ¬ 𝜓). It is traditional to use the singular form "is", not the plural form "are", in the generic expressions. By convention the major premise is listed first. In traditional Aristotelian syllogisms the predicates have a restricted form ("x is a ..."); those predicates could be modeled in modern notation by more specific constructs such as 𝑥 = 𝐴, 𝑥 ∈ 𝐴, or 𝑥 ⊆ 𝐴. Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristotelian logic are the forerunners of predicate calculus. If we used restricted forms like 𝑥 ∈ 𝐴 instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear. Using such specific constructs would also be anti-historical; Aristotle and others who directly followed his work focused on relating wholes to their parts, an approach now called part-whole theory. The work of Cantor and Peano (over 2,000 years later) led to a sharper distinction between inclusion (⊆) and membership (∈); this distinction was not directly made in Aristotle's work. There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2567, celaront 2568, cesaro 2573, camestros 2574, felapton 2579, darapti 2580, calemos 2584, fesapo 2585, and bamalip 2586. These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically, Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here. Aristotelian logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus.
Theorem	barbara 2563	"Barbara", one of the fundamental syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore all 𝜒 is 𝜓. (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.) For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ 𝑀) (all men are mortal) and ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝐻) (Socrates is a man) therefore ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝑀) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 17. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1820. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    ⇒   ⊢ ∀𝑥(𝜒 → 𝜓)
Theorem	celarent 2564	"Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)
Theorem	darii 2565	"Darii", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferio 2566	"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	barbari 2567	"Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	celaront 2568	"Celaront", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	cesare 2569	"Cesare", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2564. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜓)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	camestres 2570	"Camestres", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. (In Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	festino 2571	"Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	baroco 2572	"Baroco", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is not 𝜓, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	cesaro 2573	"Cesaro", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜓)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	camestros 2574	"Camestros", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, no 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	datisi 2575	"Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	disamis 2576	"Disamis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferison 2577	"Ferison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	bocardo 2578	"Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.) For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". A reorder of disamis 2576; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	felapton 2579	"Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	darapti 2580	"Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	calemes 2581	"Calemes", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜓 is 𝜒, therefore no 𝜒 is 𝜑. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	dimatis 2582	"Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2565 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
Theorem	fresison 2583	"Fresison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓 (PeM), and some 𝜓 is 𝜒 (MiS), therefore some 𝜒 is not 𝜑 (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜓 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	calemos 2584	"Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	fesapo 2585	"Fesapo", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜓 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜓    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	bamalip 2586	"Bamalip", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜑 exist, therefore some 𝜒 is 𝜑. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2567. (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
1.7.2  Intuitionistic logic Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 431 or peirce 193. We mostly treat intuitionistic logic in a separate file, iset.mm, which is known as the Intuitionistic Logic Explorer on the web site. However, iset.mm has a number of additional axioms (mainly to replace definitions like df-or 385 and df-ex 1705 which are not valid in intuitionistic logic) and we want to prove those axioms here to demonstrate that adding those axioms in iset.mm does not make iset.mm any less consistent than set.mm. The following axioms are unchanged between set.mm and iset.mm: ax-1 6, ax-2 7, ax-mp 5, ax-4 1737, ax-11 2034, ax-gen 1722, ax-7 1935, ax-12 2047, ax-8 1992, ax-9 1999, and ax-5 1839. In this list of axioms, the ones that repeat earlier theorems are marked "(New usage is discouraged.)" so that the earlier theorems will be used consistently in other proofs.
Theorem	axia1 2587	Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜑)
Theorem	axia2 2588	Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜓)
Theorem	axia3 2589	'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
Theorem	axin1 2590	'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Theorem	axin2 2591	'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (¬ 𝜑 → (𝜑 → 𝜓))
Theorem	axio 2592	Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
Theorem	axi4 2593	Specialization (intuitionistic logic axiom ax-4). This is just sp 2053 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	axi5r 2594	Converse of ax-c4 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))
Theorem	axial 2595	The setvar 𝑥 is not free in ∀𝑥𝜑 (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	axie1 2596	The setvar 𝑥 is not free in ∃𝑥𝜑 (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	axie2 2597	A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.)
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	axi9 2598	Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1888 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	axi10 2599	Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2307 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axi12 2600	Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2302 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. (Contributed by Jim Kingdon, 31-Dec-2017.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))