P. 22 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2101-2200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eqtri 2101	An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 = 𝐶
Theorem	eqtr2i 2102	An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐶 = 𝐴
Theorem	eqtr3i 2103	An equality transitivity inference. (Contributed by NM, 6-May-1994.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    ⇒   ⊢ 𝐵 = 𝐶
Theorem	eqtr4i 2104	An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 = 𝐶
Theorem	3eqtri 2105	An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐴 = 𝐷
Theorem	3eqtrri 2106	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐷 = 𝐴
Theorem	3eqtr2i 2107	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐴 = 𝐷
Theorem	3eqtr2ri 2108	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐷 = 𝐴
Theorem	3eqtr3i 2109	An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶 = 𝐷
Theorem	3eqtr3ri 2110	An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐷 = 𝐶
Theorem	3eqtr4i 2111	An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶 = 𝐷
Theorem	3eqtr4ri 2112	An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐷 = 𝐶
Theorem	eqtrd 2113	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	eqtr2d 2114	An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	eqtr3d 2115	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	eqtr4d 2116	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	3eqtrd 2117	A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐷)
Theorem	3eqtrrd 2118	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐴)
Theorem	3eqtr2d 2119	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐷)
Theorem	3eqtr2rd 2120	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐴)
Theorem	3eqtr3d 2121	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr3rd 2122	A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐶)
Theorem	3eqtr4d 2123	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr4rd 2124	A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐶)
Theorem	syl5eq 2125	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	syl5req 2126	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	syl5eqr 2127	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	syl5reqr 2128	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	syl6eq 2129	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	syl6req 2130	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	syl6eqr 2131	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	syl6reqr 2132	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	sylan9eq 2133	An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
Theorem	sylan9req 2134	An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ (𝜓 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
Theorem	sylan9eqr 2135	An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝐴 = 𝐶)
Theorem	3eqtr3g 2136	A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr3a 2137	A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr4g 2138	A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr4a 2139	A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	eq2tri 2140	A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹)    &   ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺)    ⇒   ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))
Theorem	eleq1 2141	Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
Theorem	eleq2 2142	Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
Theorem	eleq12 2143	Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
Theorem	eleq1i 2144	Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)
Theorem	eleq2i 2145	Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)
Theorem	eleq12i 2146	Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)
Theorem	eleq1d 2147	Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
Theorem	eleq2d 2148	Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
Theorem	eleq12d 2149	Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
Theorem	eleq1a 2150	A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))
Theorem	eqeltri 2151	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eqeltrri 2152	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ 𝐵 ∈ 𝐶
Theorem	eleqtri 2153	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eleqtrri 2154	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eqeltrd 2155	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrrd 2156	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝐶)
Theorem	eleqtrd 2157	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrrd 2158	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	3eltr3i 2159	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶 ∈ 𝐷
Theorem	3eltr4i 2160	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶 ∈ 𝐷
Theorem	3eltr3d 2161	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr4d 2162	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr3g 2163	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr4g 2164	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	syl5eqel 2165	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	syl5eqelr 2166	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	syl5eleq 2167	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	syl5eleqr 2168	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	syl6eqel 2169	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	syl6eqelr 2170	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	syl6eleq 2171	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	syl6eleqr 2172	A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleq2s 2173	Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝐴 ∈ 𝐵 → 𝜑)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝐴 ∈ 𝐶 → 𝜑)
Theorem	eqneltrd 2174	If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
Theorem	eqneltrrd 2175	If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)
Theorem	neleqtrd 2176	If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
Theorem	neleqtrrd 2177	If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
Theorem	cleqh 2178*	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2242. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    &   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	nelneq 2179	A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵)
Theorem	nelneq2 2180	A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶)
Theorem	eqsb3lem 2181*	Lemma for eqsb3 2182. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴)
Theorem	eqsb3 2182*	Substitution applied to an atomic wff (class version of equsb3 1866). (Contributed by Rodolfo Medina, 28-Apr-2010.)
⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴)
Theorem	clelsb3 2183*	Substitution applied to an atomic wff (class version of elsb3 1893). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
Theorem	clelsb4 2184*	Substitution applied to an atomic wff (class version of elsb4 1894). (Contributed by Jim Kingdon, 22-Nov-2018.)
⊢ ([𝑥 / 𝑦]𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)
Theorem	hbxfreq 2185	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1401 for equivalence version. (Contributed by NM, 21-Aug-2007.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	hblem 2186*	Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
Theorem	abeq2 2187*	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2192 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable 𝜑 (that has a free variable 𝑥) to a theorem with a class variable 𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥 ∣ 𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new set and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
Theorem	abeq1 2188*	Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴))
Theorem	abeq2i 2189	Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
⊢ 𝐴 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
Theorem	abeq1i 2190	Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
⊢ {𝑥 ∣ 𝜑} = 𝐴    ⇒   ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)
Theorem	abeq2d 2191	Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
Theorem	abbi 2192	Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
Theorem	abbi2i 2193*	Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)    ⇒   ⊢ 𝐴 = {𝑥 ∣ 𝜑}
Theorem	abbii 2194	Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}
Theorem	abbid 2195	Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
Theorem	abbidv 2196*	Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
Theorem	abbi2dv 2197*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
Theorem	abbi1dv 2198*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)
Theorem	abid2 2199*	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
Theorem	sb8ab 2200	Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}