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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqcomd 2301 Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  B  =  A )
 
Theoremeqeq1 2302 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
 
Theoremeqeq1i 2303 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  =  C 
 <->  B  =  C )
 
Theoremeqeq1d 2304 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  =  C  <->  B  =  C ) )
 
Theoremeqeq2 2305 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  =  A  <->  C  =  B ) )
 
Theoremeqeq2i 2306 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  =  A 
 <->  C  =  B )
 
Theoremeqeq2d 2307 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  =  A  <->  C  =  B ) )
 
Theoremeqeq12 2308 Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C 
 <->  B  =  D ) )
 
Theoremeqeq12i 2309 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  =  C  <->  B  =  D )
 
Theoremeqeq12d 2310 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqeqan12d 2311 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqeqan12rd 2312 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ps 
 /\  ph )  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqtr 2313 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 2314 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 2315 A transitive law for class equality. (Contributed by NM, 20-May-2005.)
 |-  ( ( A  =  C  /\  B  =  C )  ->  A  =  B )
 
Theoremeqtri 2316 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B  =  C   =>    |-  A  =  C
 
Theoremeqtr2i 2317 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
 |-  A  =  B   &    |-  B  =  C   =>    |-  C  =  A
 
Theoremeqtr3i 2318 An equality transitivity inference. (Contributed by NM, 6-May-1994.)
 |-  A  =  B   &    |-  A  =  C   =>    |-  B  =  C
 
Theoremeqtr4i 2319 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  C  =  B   =>    |-  A  =  C
 
Theorem3eqtri 2320 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
 |-  A  =  B   &    |-  B  =  C   &    |-  C  =  D   =>    |-  A  =  D
 
Theorem3eqtrri 2321 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 2322 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
 |-  A  =  B   &    |-  C  =  B   &    |-  C  =  D   =>    |-  A  =  D
 
Theorem3eqtr2ri 2323 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 2324 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 2325 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
 |-  A  =  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  D  =  C
 
Theorem3eqtr4i 2326 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 2327 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 2328 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremeqtr2d 2329 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremeqtr3d 2330 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B  =  C )
 
Theoremeqtr4d 2331 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =  C )
 
Theorem3eqtrd 2332 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 2333 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 2334 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 2335 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 2336 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 2337 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 2338 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 2339 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 2340 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl5req 2341 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  A  =  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl5eqr 2342 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  B  =  A   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl5reqr 2343 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  B  =  A   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl6eq 2344 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl6req 2345 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   &    |-  B  =  C   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl6eqr 2346 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl6reqr 2347 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   &    |-  C  =  B   =>    |-  ( ph  ->  C  =  A )
 
Theoremsylan9eq 2348 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 2349 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
 |-  ( ph  ->  B  =  A )   &    |-  ( ps  ->  B  =  C )   =>    |-  ( ( ph  /\ 
 ps )  ->  A  =  C )
 
Theoremsylan9eqr 2350 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  B  =  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  =  C )
 
Theorem3eqtr3g 2351 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 2352 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 2353 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 2354 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 2355 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 2356 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
 
Theoremeleq2 2357 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  e.  A  <->  C  e.  B ) )
 
Theoremeleq12 2358 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  e.  C 
 <->  B  e.  D ) )
 
Theoremeleq1i 2359 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  e.  C 
 <->  B  e.  C )
 
Theoremeleq2i 2360 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  e.  A 
 <->  C  e.  B )
 
Theoremeleq12i 2361 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  e.  C  <->  B  e.  D )
 
Theoremeleq1d 2362 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  e.  C  <->  B  e.  C ) )
 
Theoremeleq2d 2363 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  e.  A  <->  C  e.  B ) )
 
Theoremeleq12d 2364 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 2365 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
 |-  ( A  e.  B  ->  ( C  =  A  ->  C  e.  B ) )
 
Theoremeqeltri 2366 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B  e.  C   =>    |-  A  e.  C
 
Theoremeqeltrri 2367 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  A  e.  C   =>    |-  B  e.  C
 
Theoremeleqtri 2368 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  B   &    |-  B  =  C   =>    |-  A  e.  C
 
Theoremeleqtrri 2369 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  B   &    |-  C  =  B   =>    |-  A  e.  C
 
Theoremeqeltrd 2370 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 2371 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 2372 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 2373 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 2374 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 2375 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 2376 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 2377 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 2378 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 2379 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 2380 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  A  =  B   &    |-  ( ph  ->  B  e.  C )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl5eqelr 2381 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  B  =  A   &    |-  ( ph  ->  B  e.  C )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl5eleq 2382 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  A  e.  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl5eleqr 2383 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  A  e.  B   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl6eqel 2384 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  B  e.  C   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl6eqelr 2385 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  B  =  A )   &    |-  B  e.  C   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl6eleq 2386 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  A  e.  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  e.  C )
 
Theoremsyl6eleqr 2387 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
 |-  ( ph  ->  A  e.  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  e.  C )
 
Theoremeleq2s 2388 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 2389 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 2390 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 2391 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 2392 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 2393* 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 2394 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
 |-  ( ( A  e.  C  /\  -.  B  e.  C )  ->  -.  A  =  B )
 
Theoremnelneq2 2395 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
 |-  ( ( A  e.  B  /\  -.  A  e.  C )  ->  -.  B  =  C )
 
Theoremeqsb3lem 2396* Lemma for eqsb3 2397. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
 
Theoremeqsb3 2397* Substitution applied to an atomic wff (class version of equsb3 2054). (Contributed by Rodolfo Medina, 28-Apr-2010.)
 |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
 
Theoremclelsb3 2398* Substitution applied to an atomic wff (class version of elsb3 2055). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
 
Theoremhbxfreq 2399 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1558 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 2400* Change the free variable of a hypothesis builder. Lemma for nfcrii 2425. (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 )
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