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Theorem psseq12d 3038
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 3036 . 2  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  C ) )
3 psseq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43psseq2d 3037 . 2  |-  ( ph  ->  ( B  C.  C  <->  B 
C.  D ) )
52, 4bitrd 177 1  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    C. wpss 2918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933
This theorem is referenced by: (None)
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