P. 17 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1601-1700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nf4r 1601	If ph is always true or always false, then variable x is effectively not free in ph. The converse holds given a decidability condition, as seen at nf4dc 1600. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ( ( A. x ph \/ A. x -. ph ) -> F/ x ph )
Theorem	19.36i 1602	Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- F/ x ps   &    |- E. x ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	19.36-1 1603	Closed form of 19.36i 1602. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)
|- F/ x ps   =>    |- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) )
Theorem	19.37-1 1604	One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)
|- F/ x ph   =>    |- ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) )
Theorem	19.37aiv 1605*	Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- E. x ( ph -> ps )   =>    |- ( ph -> E. x ps )
Theorem	19.38 1606	Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
Theorem	19.23t 1607	Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
|- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	19.23 1608	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- F/ x ps   =>    |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	19.32dc 1609	Theorem 19.32 of [Margaris] p. 90, where ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
|- F/ x ph   =>    |- (DECID ph -> ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) )
Theorem	19.32r 1610	One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if ph is decidable, as seen at 19.32dc 1609. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- F/ x ph   =>    |- ( ( ph \/ A. x ps ) -> A. x ( ph \/ ps ) )
Theorem	19.31r 1611	One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- F/ x ps   =>    |- ( ( A. x ph \/ ps ) -> A. x ( ph \/ ps ) )
Theorem	19.44 1612	Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ps   =>    |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ ps ) )
Theorem	19.45 1613	Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ph   =>    |- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) )
Theorem	19.34 1614	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) )
Theorem	19.41h 1615	Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1616 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
|- ( ps -> A. x ps )   =>    |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )
Theorem	19.41 1616	Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
|- F/ x ps   =>    |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )
Theorem	19.42h 1617	Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1618 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
|- ( ph -> A. x ph )   =>    |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
Theorem	19.42 1618	Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
|- F/ x ph   =>    |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
Theorem	excom13 1619	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. z E. y E. x ph )
Theorem	exrot3 1620	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. y E. z E. x ph )
Theorem	exrot4 1621	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z E. w ph <-> E. z E. w E. x E. y ph )
Theorem	nexr 1622	Inference from 19.8a 1522. (Contributed by Jeff Hankins, 26-Jul-2009.)
|- -. E. x ph   =>    |- -. ph
Theorem	exan 1623	Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( E. x ph /\ ps )   =>    |- E. x ( ph /\ ps )
Theorem	hbexd 1624	Deduction form of bound-variable hypothesis builder hbex 1567. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( E. y ps -> A. x E. y ps ) )
Theorem	eeor 1625	Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) )
1.3.8  Equality theorems without distinct variables
Theorem	a9e 1626	At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1376 through ax-14 1445 and ax-17 1459, all axioms other than ax-9 1464 are believed to be theorems of free logic, although the system without ax-9 1464 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- E. x x = y
Theorem	a9ev 1627*	At least one individual exists. Weaker version of a9e 1626. (Contributed by NM, 3-Aug-2017.)
|- E. x x = y
Theorem	ax9o 1628	An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( A. x ( x = y -> A. x ph ) -> ph )
Theorem	equid 1629	Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms. This proof is similar to Tarski's and makes use of a dummy variable y. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)
|- x = x
Theorem	nfequid 1630	Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
|- F/ y x = x
Theorem	stdpc6 1631	One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1693.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
|- A. x x = x
Theorem	equcomi 1632	Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> y = x )
Theorem	equcom 1633	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
|- ( x = y <-> y = x )
Theorem	equcoms 1634	An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ph )   =>    |- ( y = x -> ph )
Theorem	equtr 1635	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( y = z -> x = z ) )
Theorem	equtrr 1636	A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( z = x -> z = y ) )
Theorem	equtr2 1637	A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ( x = z /\ y = z ) -> x = y )
Theorem	equequ1 1638	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x = z <-> y = z ) )
Theorem	equequ2 1639	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z = x <-> z = y ) )
Theorem	elequ1 1640	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x e. z <-> y e. z ) )
Theorem	elequ2 1641	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z e. x <-> z e. y ) )
Theorem	ax11i 1642	Inference that has ax-11 1437 (without A. y) as its conclusion and doesn't require ax-10 1436, ax-11 1437, or ax-12 1442 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( ps -> A. x ps )   =>    |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
1.3.9  Axioms ax-10 and ax-11
Theorem	ax10o 1643	Show that ax-10o 1644 can be derived from ax-10 1436. An open problem is whether this theorem can be derived from ax-10 1436 and the others when ax-11 1437 is replaced with ax-11o 1744. See theorem ax10 1645 for the rederivation of ax-10 1436 from ax10o 1643. Normally, ax10o 1643 should be used rather than ax-10o 1644, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)
|- ( A. x x = y -> ( A. x ph -> A. y ph ) )
Axiom	ax-10o 1644	Axiom ax-10o 1644 ("o" for "old") was the original version of ax-10 1436, before it was discovered (in May 2008) that the shorter ax-10 1436 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint). This axiom is redundant, as shown by theorem ax10o 1643. Normally, ax10o 1643 should be used rather than ax-10o 1644, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( A. x x = y -> ( A. x ph -> A. y ph ) )
Theorem	ax10 1645	Rederivation of ax-10 1436 from original version ax-10o 1644. See theorem ax10o 1643 for the derivation of ax-10o 1644 from ax-10 1436. This theorem should not be referenced in any proof. Instead, use ax-10 1436 above so that uses of ax-10 1436 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
|- ( A. x x = y -> A. y y = x )
Theorem	hbae 1646	All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( A. x x = y -> A. z A. x x = y )
Theorem	nfae 1647	All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ z A. x x = y
Theorem	hbaes 1648	Rule that applies hbae 1646 to antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. z A. x x = y -> ph )   =>    |- ( A. x x = y -> ph )
Theorem	hbnae 1649	All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
|- ( -. A. x x = y -> A. z -. A. x x = y )
Theorem	nfnae 1650	All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ z -. A. x x = y
Theorem	hbnaes 1651	Rule that applies hbnae 1649 to antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. z -. A. x x = y -> ph )   =>    |- ( -. A. x x = y -> ph )
Theorem	naecoms 1652	A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ph )   =>    |- ( -. A. y y = x -> ph )
Theorem	equs4 1653	Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
|- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
Theorem	equsalh 1654	A useful equivalence related to substitution. New proofs should use equsal 1655 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ( x = y -> ph ) <-> ps )
Theorem	equsal 1655	A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ( x = y -> ph ) <-> ps )
Theorem	equsex 1656	A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ( x = y /\ ph ) <-> ps )
Theorem	equsexd 1657	Deduction form of equsex 1656. (Contributed by Jim Kingdon, 29-Dec-2017.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( E. x ( x = y /\ ps ) <-> ch ) )
Theorem	dral1 1658	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )
Theorem	dral2 1659	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )
Theorem	drex2 1660	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( E. z ph <-> E. z ps ) )
Theorem	drnf1 1661	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) )
Theorem	drnf2 1662	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) )
Theorem	spimth 1663	Closed theorem form of spim 1666. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)
|- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )
Theorem	spimt 1664	Closed theorem form of spim 1666. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
|- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )
Theorem	spimh 1665	Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1666 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spim 1666	Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1666 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spimeh 1667	Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)
|- ( ph -> A. x ph )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	spimed 1668	Deduction version of spime 1669. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
|- ( ch -> F/ x ph )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ch -> ( ph -> E. x ps ) )
Theorem	spime 1669	Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
|- F/ x ph   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	cbv3 1670	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv3h 1671	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
|- ( ph -> A. y ph )   &    |- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv1 1672	Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> A. y ch ) )
Theorem	cbv1h 1673	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
|- ( ph -> ( ps -> A. y ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
Theorem	cbv2h 1674	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps -> A. y ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )
Theorem	cbv2 1675	Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( A. x ps <-> A. y ch ) )
Theorem	cbvalh 1676	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A. y ph )   &    |- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbval 1677	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbvexh 1678	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)
|- ( ph -> A. y ph )   &    |- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	cbvex 1679	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	chvar 1680	Implicit substitution of y for x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	equvini 1681	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require z to be distinct from x and y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( x = y -> E. z ( x = z /\ z = y ) )
Theorem	equveli 1682	A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1681.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
|- ( A. z ( z = x <-> z = y ) -> x = y )
Theorem	nfald 1683	If x is not free in ph, it is not free in A. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x A. y ps )
Theorem	nfexd 1684	If x is not free in ph, it is not free in E. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E. y ps )
1.3.10  Substitution (without distinct variables)
Syntax	wsb 1685	Extend wff definition to include proper substitution (read "the wff that results when y is properly substituted for x in wff ph"). (Contributed by NM, 24-Jan-2006.)
wff [ y / x ] ph
Definition	df-sb 1686	Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [ y / x ] ph to mean "the wff that results when y is properly substituted for x in the wff ph." We can also use [ y / x ] ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1698. Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " ph ( y ) is the wff that results when y is properly substituted for x in ph ( x )." For example, if the original ph ( x ) is x = y, then ph ( y ) is y = y, from which we obtain that ph ( x ) is x = x. So what exactly does ph ( x ) mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1761, sbcom2 1904 and sbid2v 1913). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1697 shows. We achieve this by having x free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 1908 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1911. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1808 and sb6 1807. In classical logic, another possible definition is ( x = y /\ ph ) \/ A. x ( x = y -> ph ) but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff ph. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
Theorem	sbimi 1687	Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
|- ( ph -> ps )   =>    |- ( [ y / x ] ph -> [ y / x ] ps )
Theorem	sbbii 1688	Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- ( [ y / x ] ph <-> [ y / x ] ps )
Theorem	sb1 1689	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
Theorem	sb2 1690	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
Theorem	sbequ1 1691	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> [ y / x ] ph ) )
Theorem	sbequ2 1692	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )
Theorem	stdpc7 1693	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1631.) Translated to traditional notation, it can be read: " x = y -> ( ph ( x, x ) -> ph ( x, y ) ), provided that y is free for x in ph ( x, y )." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
|- ( x = y -> ( [ x / y ] ph -> ph ) )
Theorem	sbequ12 1694	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph <-> [ y / x ] ph ) )
Theorem	sbequ12r 1695	An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( x = y -> ( [ x / y ] ph <-> ph ) )
Theorem	sbequ12a 1696	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
Theorem	sbid 1697	An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)
|- ( [ x / x ] ph <-> ph )
Theorem	stdpc4 1698	The specialization axiom of standard predicate calculus. It states that if a statement ph holds for all x, then it also holds for the specific case of y (properly) substituted for x. Translated to traditional notation, it can be read: " A. x ph ( x ) -> ph ( y ), provided that y is free for x in ph ( x )." Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ph -> [ y / x ] ph )
Theorem	sbh 1699	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] ph <-> ph )
Theorem	sbf 1700	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
|- F/ x ph   =>    |- ( [ y / x ] ph <-> ph )