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Theorem psseq1d 3431
Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
Hypothesis
Ref Expression
psseq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
psseq1d  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  C ) )

Proof of Theorem psseq1d
StepHypRef Expression
1 psseq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 psseq1 3426 . 2  |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C ) )
31, 2syl 16 1  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    C. wpss 3313
This theorem is referenced by:  psseq12d  3433  fin23lem32  8216  fin23lem35  8219  compssiso  8246  mrieqv2d  13856  mrissmrcd  13857  pgpfac1lem5  15629  islbs3  16219  chpsscon2  22999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-ne 2600  df-in 3319  df-ss 3326  df-pss 3328
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