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Theorem psseq2 3277
Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
psseq2  |-  ( A  =  B  ->  ( C  C.  A  <->  C  C.  B ) )

Proof of Theorem psseq2
StepHypRef Expression
1 sseq2 3213 . . 3  |-  ( A  =  B  ->  ( C  C_  A  <->  C  C_  B
) )
2 neeq2 2468 . . 3  |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
31, 2anbi12d 691 . 2  |-  ( A  =  B  ->  (
( C  C_  A  /\  C  =/=  A
)  <->  ( C  C_  B  /\  C  =/=  B
) ) )
4 df-pss 3181 . 2  |-  ( C 
C.  A  <->  ( C  C_  A  /\  C  =/= 
A ) )
5 df-pss 3181 . 2  |-  ( C 
C.  B  <->  ( C  C_  B  /\  C  =/= 
B ) )
63, 4, 53bitr4g 279 1  |-  ( A  =  B  ->  ( C  C.  A  <->  C  C.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    =/= wne 2459    C_ wss 3165    C. wpss 3166
This theorem is referenced by:  psseq2i  3279  psseq2d  3282  psssstr  3295  brrpssg  6295  sorpssint  6303  php  7061  php2  7062  pssnn  7097  isfin4  7939  fin2i2  7960  elnp  8627  elnpi  8628  ltprord  8670  pgpfac1lem1  15325  pgpfac1lem5  15330  lbsextlem4  15930  alexsubALTlem4  17760  spansncv  22248  cvbr  22878  cvcon3  22880  cvnbtwn  22882  cvbr4i  22963  dfon2lem6  24215  dfon2lem7  24216  dfon2lem8  24217  dfon2  24219  lcvbr  29833  lcvnbtwn  29837  lsatcv0  29843  lsat0cv  29845  islshpcv  29865  mapdcv  32472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-in 3172  df-ss 3179  df-pss 3181
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