MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psseq12d Structured version   Unicode version

Theorem psseq12d 3433
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 3431 . 2  |-  ( ph  ->  ( A  C.  C  <->  B 
C.  C ) )
3 psseq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43psseq2d 3432 . 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 1652    C. wpss 3313
This theorem is referenced by:  fin23lem32  8216  fin23lem34  8218  fin23lem35  8219  fin23lem41  8224  isf32lem5  8229  isf32lem6  8230  isf32lem11  8235  compssiso  8246  canthp1lem2  8520  chnle  23008
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
  Copyright terms: Public domain W3C validator