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Theorem psseq1 3276
Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
psseq1  |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C ) )

Proof of Theorem psseq1
StepHypRef Expression
1 sseq1 3212 . . 3  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
2 neeq1 2467 . . 3  |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
31, 2anbi12d 691 . 2  |-  ( A  =  B  ->  (
( A  C_  C  /\  A  =/=  C
)  <->  ( B  C_  C  /\  B  =/=  C
) ) )
4 df-pss 3181 . 2  |-  ( A 
C.  C  <->  ( A  C_  C  /\  A  =/= 
C ) )
5 df-pss 3181 . 2  |-  ( B 
C.  C  <->  ( B  C_  C  /\  B  =/= 
C ) )
63, 4, 53bitr4g 279 1  |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C ) )
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:  psseq1i  3278  psseq1d  3281  psstr  3293  sspsstr  3294  brrpssg  6295  sorpssuni  6302  pssnn  7097  marypha1lem  7202  infeq5i  7353  infpss  7859  fin4i  7940  isfin2-2  7961  zornn0g  8148  ttukeylem7  8158  elnp  8627  elnpi  8628  ltprord  8670  pgpfac1lem1  15325  pgpfac1lem5  15330  pgpfac1  15331  pgpfaclem2  15333  pgpfac  15335  islbs3  15924  alexsubALTlem4  17760  wilthlem2  20323  spansncv  22248  cvbr  22878  cvcon3  22880  cvnbtwn  22882  dfon2lem3  24212  dfon2lem4  24213  dfon2lem5  24214  dfon2lem6  24215  dfon2lem7  24216  dfon2lem8  24217  dfon2  24219  lcvbr  29833  lcvnbtwn  29837  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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