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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqtr 2301 Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
 |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )
 
Theoremeqtr2 2302 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( A  =  B  /\  A  =  C )  ->  B  =  C )
 
Theoremeqtr3 2303 A transitive law for class equality. (Contributed by NM, 20-May-2005.)
 |-  ( ( A  =  C  /\  B  =  C )  ->  A  =  B )
 
Theoremeqtri 2304 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B  =  C   =>    |-  A  =  C
 
Theoremeqtr2i 2305 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
 |-  A  =  B   &    |-  B  =  C   =>    |-  C  =  A
 
Theoremeqtr3i 2306 An equality transitivity inference. (Contributed by NM, 6-May-1994.)
 |-  A  =  B   &    |-  A  =  C   =>    |-  B  =  C
 
Theoremeqtr4i 2307 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  C  =  B   =>    |-  A  =  C
 
Theorem3eqtri 2308 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
 |-  A  =  B   &    |-  B  =  C   &    |-  C  =  D   =>    |-  A  =  D
 
Theorem3eqtrri 2309 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 2310 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
 |-  A  =  B   &    |-  C  =  B   &    |-  C  =  D   =>    |-  A  =  D
 
Theorem3eqtr2ri 2311 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 2312 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 2313 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
 |-  A  =  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  D  =  C
 
Theorem3eqtr4i 2314 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 2315 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 2316 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremeqtr2d 2317 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremeqtr3d 2318 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B  =  C )
 
Theoremeqtr4d 2319 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =  C )
 
Theorem3eqtrd 2320 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  A  =  D )
 
Theorem3eqtrrd 2321 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  D  =  A )
 
Theorem3eqtr2d 2322 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  A  =  D )
 
Theorem3eqtr2rd 2323 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  D  =  A )
 
Theorem3eqtr3d 2324 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr3rd 2325 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  D  =  C )
 
Theorem3eqtr4d 2326 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr4rd 2327 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  D  =  C )
 
Theoremsyl5eq 2328 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl5req 2329 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  A  =  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl5eqr 2330 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  B  =  A   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl5reqr 2331 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  B  =  A   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl6eq 2332 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl6req 2333 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   &    |-  B  =  C   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl6eqr 2334 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl6reqr 2335 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   &    |-  C  =  B   =>    |-  ( ph  ->  C  =  A )
 
Theoremsylan9eq 2336 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  B  =  C )   =>    |-  ( ( ph  /\ 
 ps )  ->  A  =  C )
 
Theoremsylan9req 2337 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
 |-  ( ph  ->  B  =  A )   &    |-  ( ps  ->  B  =  C )   =>    |-  ( ( ph  /\ 
 ps )  ->  A  =  C )
 
Theoremsylan9eqr 2338 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  B  =  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  =  C )
 
Theorem3eqtr3g 2339 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  A  =  C   &    |-  B  =  D   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr3a 2340 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |-  A  =  B   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr4g 2341 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  C  =  A   &    |-  D  =  B   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr4a 2342 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   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  =  D )
 
Theoremeq2tri 2343 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 ) )
 
Theoremeleq1 2344 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
 
Theoremeleq2 2345 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  e.  A  <->  C  e.  B ) )
 
Theoremeleq12 2346 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  e.  C 
 <->  B  e.  D ) )
 
Theoremeleq1i 2347 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  e.  C 
 <->  B  e.  C )
 
Theoremeleq2i 2348 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  e.  A 
 <->  C  e.  B )
 
Theoremeleq12i 2349 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  e.  C  <->  B  e.  D )
 
Theoremeleq1d 2350 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  e.  C  <->  B  e.  C ) )
 
Theoremeleq2d 2351 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  e.  A  <->  C  e.  B ) )
 
Theoremeleq12d 2352 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  e.  C  <->  B  e.  D ) )
 
Theoremeleq1a 2353 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
 |-  ( A  e.  B  ->  ( C  =  A  ->  C  e.  B ) )
 
Theoremeqeltri 2354 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B  e.  C   =>    |-  A  e.  C
 
Theoremeqeltrri 2355 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  A  e.  C   =>    |-  B  e.  C
 
Theoremeleqtri 2356 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  B   &    |-  B  =  C   =>    |-  A  e.  C
 
Theoremeleqtrri 2357 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  B   &    |-  C  =  B   =>    |-  A  e.  C
 
Theoremeqeltrd 2358 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  e.  C )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremeqeltrrd 2359 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  e.  C )   =>    |-  ( ph  ->  B  e.  C )
 
Theoremeleqtrd 2360 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremeleqtrrd 2361 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  e.  C )
 
Theorem3eltr3i 2362 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  A  e.  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  e.  D
 
Theorem3eltr4i 2363 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  A  e.  B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  C  e.  D
 
Theorem3eltr3d 2364 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  e.  D )
 
Theorem3eltr4d 2365 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  e.  D )
 
Theorem3eltr3g 2366 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  A  =  C   &    |-  B  =  D   =>    |-  ( ph  ->  C  e.  D )
 
Theorem3eltr4g 2367 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  C  =  A   &    |-  D  =  B   =>    |-  ( ph  ->  C  e.  D )
 
Theoremsyl5eqel 2368 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  A  =  B   &    |-  ( ph  ->  B  e.  C )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl5eqelr 2369 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  B  =  A   &    |-  ( ph  ->  B  e.  C )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl5eleq 2370 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  A  e.  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl5eleqr 2371 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  A  e.  B   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl6eqel 2372 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  B  e.  C   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl6eqelr 2373 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  B  =  A )   &    |-  B  e.  C   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl6eleq 2374 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  A  e.  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl6eleqr 2375 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
 |-  ( ph  ->  A  e.  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  e.  C )
 
Theoremeleq2s 2376 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( A  e.  B  -> 
 ph )   &    |-  C  =  B   =>    |-  ( A  e.  C  ->  ph )
 
Theoremeqneltrd 2377 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.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  -.  B  e.  C )   =>    |-  ( ph  ->  -.  A  e.  C )
 
Theoremeqneltrrd 2378 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.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  -.  A  e.  C )   =>    |-  ( ph  ->  -.  B  e.  C )
 
Theoremneleqtrd 2379 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.)
 |-  ( ph  ->  -.  C  e.  A )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -.  C  e.  B )
 
Theoremneleqtrrd 2380 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.)
 |-  ( ph  ->  -.  C  e.  B )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -.  C  e.  A )
 
Theoremcleqh 2381* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.)
 |-  ( y  e.  A  ->  A. x  y  e.  A )   &    |-  ( y  e.  B  ->  A. x  y  e.  B )   =>    |-  ( A  =  B 
 <-> 
 A. x ( x  e.  A  <->  x  e.  B ) )
 
Theoremnelneq 2382 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
 |-  ( ( A  e.  C  /\  -.  B  e.  C )  ->  -.  A  =  B )
 
Theoremnelneq2 2383 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
 |-  ( ( A  e.  B  /\  -.  A  e.  C )  ->  -.  B  =  C )
 
Theoremeqsb3lem 2384* Lemma for eqsb3 2385. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
 
Theoremeqsb3 2385* Substitution applied to an atomic wff (class version of equsb3 2041). (Contributed by Rodolfo Medina, 28-Apr-2010.)
 |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
 
Theoremclelsb3 2386* Substitution applied to an atomic wff (class version of elsb3 2042). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
 
Theoremhbxfreq 2387 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1556 for equivalence version. (Contributed by NM, 21-Aug-2007.)
 |-  A  =  B   &    |-  (
 y  e.  B  ->  A. x  y  e.  B )   =>    |-  ( y  e.  A  ->  A. x  y  e.  A )
 
Theoremhblem 2388* Change the free variable of a hypothesis builder. Lemma for nfcrii 2413. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  ( y  e.  A  ->  A. x  y  e.  A )   =>    |-  ( z  e.  A  ->  A. x  z  e.  A )
 
Theoremabeq2 2389* 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 2394 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  ph (that has a free variable  x) to a theorem with a class variable  A, we substitute  x  e.  A for  ph throughout and simplify, where  A is a new class variable not already in the wff. An example is the conversion of zfauscl 4144 to inex1 4156 (look at the instance of zfauscl 4144 that occurs in the proof of inex1 4156). Conversely, to convert a theorem with a class variable  A to one with 
ph, we substitute  { x  | 
ph } for  A throughout and simplify, where  x and  ph are new set and wff variables not already in the wff. An example is cp 7556, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 7555. 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  |  ph }  <->  A. x ( x  e.  A  <->  ph ) )
 
Theoremabeq1 2390* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
 |-  ( { x  |  ph
 }  =  A  <->  A. x ( ph  <->  x  e.  A ) )
 
Theoremabeq2i 2391 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
 |-  A  =  { x  |  ph }   =>    |-  ( x  e.  A  <->  ph )
 
Theoremabeq1i 2392 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
 |- 
 { x  |  ph }  =  A   =>    |-  ( ph  <->  x  e.  A )
 
Theoremabeq2d 2393 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
 |-  ( ph  ->  A  =  { x  |  ps } )   =>    |-  ( ph  ->  ( x  e.  A  <->  ps ) )
 
Theoremabbi 2394 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
 |-  ( A. x (
 ph 
 <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
 
Theoremabbi2i 2395* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  <->  ph )   =>    |-  A  =  { x  |  ph }
 
Theoremabbii 2396 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |- 
 { x  |  ph }  =  { x  |  ps }
 
Theoremabbid 2397 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
 
Theoremabbidv 2398* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
 
Theoremabbi2dv 2399* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
 |-  ( ph  ->  ( x  e.  A  <->  ps ) )   =>    |-  ( ph  ->  A  =  { x  |  ps } )
 
Theoremabbi1dv 2400* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
 |-  ( ph  ->  ( ps 
 <->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  =  A )
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