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Theorem psseq12d 3377
Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
Hypotheses
Ref Expression
psseq1d.1  |-  ( ph  ->  A  =  B )
psseq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
psseq12d  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  D ) )

Proof of Theorem psseq12d
StepHypRef Expression
1 psseq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21psseq1d 3375 . 2  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  C ) )
3 psseq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43psseq2d 3376 . 2  |-  ( ph  ->  ( B  C.  C  <->  B 
C.  D ) )
52, 4bitrd 245 1  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    C. wpss 3257
This theorem is referenced by:  fin23lem32  8150  fin23lem34  8152  fin23lem35  8153  fin23lem41  8158  isf32lem5  8163  isf32lem6  8164  isf32lem11  8169  compssiso  8180  canthp1lem2  8454  chnle  22857
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
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 2367  df-cleq 2373  df-clel 2376  df-ne 2545  df-in 3263  df-ss 3270  df-pss 3272
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