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Theorem List for Intuitionistic Logic Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqtr2 2101 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
((𝐴 = 𝐵𝐴 = 𝐶) → 𝐵 = 𝐶)
 
Theoremeqtr3 2102 A transitive law for class equality. (Contributed by NM, 20-May-2005.)
((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
 
Theoremeqtri 2103 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐵 = 𝐶       𝐴 = 𝐶
 
Theoremeqtr2i 2104 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
𝐴 = 𝐵    &   𝐵 = 𝐶       𝐶 = 𝐴
 
Theoremeqtr3i 2105 An equality transitivity inference. (Contributed by NM, 6-May-1994.)
𝐴 = 𝐵    &   𝐴 = 𝐶       𝐵 = 𝐶
 
Theoremeqtr4i 2106 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐶 = 𝐵       𝐴 = 𝐶
 
Theorem3eqtri 2107 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
𝐴 = 𝐵    &   𝐵 = 𝐶    &   𝐶 = 𝐷       𝐴 = 𝐷
 
Theorem3eqtrri 2108 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
 
Theorem3eqtr2i 2109 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
𝐴 = 𝐵    &   𝐶 = 𝐵    &   𝐶 = 𝐷       𝐴 = 𝐷
 
Theorem3eqtr2ri 2110 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
 
Theorem3eqtr3i 2111 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
 
Theorem3eqtr3ri 2112 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
𝐴 = 𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐷 = 𝐶
 
Theorem3eqtr4i 2113 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
 
Theorem3eqtr4ri 2114 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
 
Theoremeqtrd 2115 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)
 
Theoremeqtr2d 2116 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)
 
Theoremeqtr3d 2117 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = 𝐶)       (𝜑𝐵 = 𝐶)
 
Theoremeqtr4d 2118 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴 = 𝐶)
 
Theorem3eqtrd 2119 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐴 = 𝐷)
 
Theorem3eqtrrd 2120 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐷 = 𝐴)
 
Theorem3eqtr2d 2121 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐴 = 𝐷)
 
Theorem3eqtr2rd 2122 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐷 = 𝐴)
 
Theorem3eqtr3d 2123 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶 = 𝐷)
 
Theorem3eqtr3rd 2124 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐷 = 𝐶)
 
Theorem3eqtr4d 2125 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶 = 𝐷)
 
Theorem3eqtr4rd 2126 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐷 = 𝐶)
 
Theoremsyl5eq 2127 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)
 
Theoremsyl5req 2128 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
𝐴 = 𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)
 
Theoremsyl5eqr 2129 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
𝐵 = 𝐴    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)
 
Theoremsyl5reqr 2130 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
𝐵 = 𝐴    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)
 
Theoremsyl6eq 2131 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)    &   𝐵 = 𝐶       (𝜑𝐴 = 𝐶)
 
Theoremsyl6req 2132 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)    &   𝐵 = 𝐶       (𝜑𝐶 = 𝐴)
 
Theoremsyl6eqr 2133 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐵       (𝜑𝐴 = 𝐶)
 
Theoremsyl6reqr 2134 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐵       (𝜑𝐶 = 𝐴)
 
Theoremsylan9eq 2135 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐵 = 𝐶)       ((𝜑𝜓) → 𝐴 = 𝐶)
 
Theoremsylan9req 2136 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
(𝜑𝐵 = 𝐴)    &   (𝜓𝐵 = 𝐶)       ((𝜑𝜓) → 𝐴 = 𝐶)
 
Theoremsylan9eqr 2137 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐵 = 𝐶)       ((𝜓𝜑) → 𝐴 = 𝐶)
 
Theorem3eqtr3g 2138 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
(𝜑𝐴 = 𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶 = 𝐷)
 
Theorem3eqtr3a 2139 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
𝐴 = 𝐵    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶 = 𝐷)
 
Theorem3eqtr4g 2140 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶 = 𝐷)
 
Theorem3eqtr4a 2141 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    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶 = 𝐷)
 
Theoremeq2tri 2142 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
(𝐴 = 𝐶𝐷 = 𝐹)    &   (𝐵 = 𝐷𝐶 = 𝐺)       ((𝐴 = 𝐶𝐵 = 𝐹) ↔ (𝐵 = 𝐷𝐴 = 𝐺))
 
Theoremeleq1w 2143 Weaker version of eleq1 2145 (but more general than elequ1 1642) not depending on ax-ext 2065 nor df-cleq 2076. (Contributed by BJ, 24-Jun-2019.)
(𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
 
Theoremeleq2w 2144 Weaker version of eleq2 2146 (but more general than elequ2 1643) not depending on ax-ext 2065 nor df-cleq 2076. (Contributed by BJ, 29-Sep-2019.)
(𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
 
Theoremeleq1 2145 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremeleq2 2146 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremeleq12 2147 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
 
Theoremeleq1i 2148 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremeleq2i 2149 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremeleq12i 2150 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)
 
Theoremeleq1d 2151 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremeleq2d 2152 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremeleq12d 2153 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremeleq1a 2154 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
(𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))
 
Theoremeqeltri 2155 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqeltrri 2156 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶
 
Theoremeleqtri 2157 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremeleqtrri 2158 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremeqeltrd 2159 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqeltrrd 2160 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)
 
Theoremeleqtrd 2161 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremeleqtrrd 2162 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theorem3eltr3i 2163 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷
 
Theorem3eltr4i 2164 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷
 
Theorem3eltr3d 2165 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)
 
Theorem3eltr4d 2166 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)
 
Theorem3eltr3g 2167 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3eltr4g 2168 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theoremsyl5eqel 2169 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴 = 𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5eqelr 2170 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5eleq 2171 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5eleqr 2172 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theoremsyl6eqel 2173 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6eqelr 2174 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6eleq 2175 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6eleqr 2176 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)
 
Theoremeleq2s 2177 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝐴𝐵𝜑)    &   𝐶 = 𝐵       (𝐴𝐶𝜑)
 
Theoremeqneltrd 2178 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.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐵𝐶)       (𝜑 → ¬ 𝐴𝐶)
 
Theoremeqneltrrd 2179 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.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐴𝐶)       (𝜑 → ¬ 𝐵𝐶)
 
Theoremneleqtrd 2180 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 → ¬ 𝐶𝐴)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐶𝐵)
 
Theoremneleqtrrd 2181 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 → ¬ 𝐶𝐵)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐶𝐴)
 
Theoremcleqh 2182* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2246. (Contributed by NM, 5-Aug-1993.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremnelneq 2183 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)
 
Theoremnelneq2 2184 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)
 
Theoremeqsb3lem 2185* Lemma for eqsb3 2186. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
 
Theoremeqsb3 2186* Substitution applied to an atomic wff (class version of equsb3 1868). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
 
Theoremclelsb3 2187* Substitution applied to an atomic wff (class version of elsb3 1895). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
 
Theoremclelsb4 2188* Substitution applied to an atomic wff (class version of elsb4 1896). (Contributed by Jim Kingdon, 22-Nov-2018.)
([𝑥 / 𝑦]𝐴𝑦𝐴𝑥)
 
Theoremhbxfreq 2189 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1402 for equivalence version. (Contributed by NM, 21-Aug-2007.)
𝐴 = 𝐵    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theoremhblem 2190* Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 
Theoremabeq2 2191* 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 2196 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 𝑥𝐴 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 {𝑥𝜑} 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 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremabeq1 2192* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
 
Theoremabeq2i 2193 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)
 
Theoremabeq1i 2194 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
{𝑥𝜑} = 𝐴       (𝜑𝑥𝐴)
 
Theoremabeq2d 2195 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
(𝜑𝐴 = {𝑥𝜓})       (𝜑 → (𝑥𝐴𝜓))
 
Theoremabbi 2196 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
(∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
 
Theoremabbi2i 2197* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}
 
Theoremabbii 2198 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}
 
Theoremabbid 2199 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theoremabbidv 2200* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
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