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Theorem psseq12d 3594
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 3592 . 2  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  C ) )
3 psseq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43psseq2d 3593 . 2  |-  ( ph  ->  ( B  C.  C  <->  B 
C.  D ) )
52, 4bitrd 253 1  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    C. wpss 3472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-ne 2654  df-in 3478  df-ss 3485  df-pss 3487
This theorem is referenced by:  fin23lem32  8741  fin23lem34  8743  fin23lem35  8744  fin23lem41  8749  isf32lem5  8754  isf32lem6  8755  isf32lem11  8760  compssiso  8771  canthp1lem2  9048  chnle  26558
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