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Theorem psseq12d 3270
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 3268 . 2  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  C ) )
3 psseq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43psseq2d 3269 . 2  |-  ( ph  ->  ( B  C.  C  <->  B 
C.  D ) )
52, 4bitrd 244 1  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    C. wpss 3153
This theorem is referenced by:  fin23lem32  7970  fin23lem34  7972  fin23lem35  7973  fin23lem41  7978  isf32lem5  7983  isf32lem6  7984  isf32lem11  7989  compssiso  8000  canthp1lem2  8275  chnle  22093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-in 3159  df-ss 3166  df-pss 3168
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