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Theorem psseq1 3394
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 3329 . . 3  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
2 neeq1 2575 . . 3  |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
31, 2anbi12d 692 . 2  |-  ( A  =  B  ->  (
( A  C_  C  /\  A  =/=  C
)  <->  ( B  C_  C  /\  B  =/=  C
) ) )
4 df-pss 3296 . 2  |-  ( A 
C.  C  <->  ( A  C_  C  /\  A  =/= 
C ) )
5 df-pss 3296 . 2  |-  ( B 
C.  C  <->  ( B  C_  C  /\  B  =/= 
C ) )
63, 4, 53bitr4g 280 1  |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    =/= wne 2567    C_ wss 3280    C. wpss 3281
This theorem is referenced by:  psseq1i  3396  psseq1d  3399  psstr  3411  sspsstr  3412  brrpssg  6483  sorpssuni  6490  pssnn  7286  marypha1lem  7396  infeq5i  7547  infpss  8053  fin4i  8134  isfin2-2  8155  zornn0g  8341  ttukeylem7  8351  elnp  8820  elnpi  8821  ltprord  8863  pgpfac1lem1  15587  pgpfac1lem5  15592  pgpfac1  15593  pgpfaclem2  15595  pgpfac  15597  islbs3  16182  alexsubALTlem4  18034  wilthlem2  20805  spansncv  23108  cvbr  23738  cvcon3  23740  cvnbtwn  23742  dfon2lem3  25355  dfon2lem4  25356  dfon2lem5  25357  dfon2lem6  25358  dfon2lem7  25359  dfon2lem8  25360  dfon2  25362  lcvbr  29504  lcvnbtwn  29508  mapdcv  32143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-ne 2569  df-in 3287  df-ss 3294  df-pss 3296
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