MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psseq2d Unicode version

Theorem psseq2d 3303
Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
Hypothesis
Ref Expression
psseq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
psseq2d  |-  ( ph  ->  ( C  C.  A  <->  C 
C.  B ) )

Proof of Theorem psseq2d
StepHypRef Expression
1 psseq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 psseq2 3298 . 2  |-  ( A  =  B  ->  ( C  C.  A  <->  C  C.  B ) )
31, 2syl 15 1  |-  ( ph  ->  ( C  C.  A  <->  C 
C.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    C. wpss 3187
This theorem is referenced by:  psseq12d  3304  php3  7090  inf3lem5  7378  infeq5i  7382  ackbij1lem15  7905  fin4en1  7980  chpsscon1  22138  chnle  22148  atcvatlem  23020  atcvati  23021  lsatcvatlem  29057  lsatcvat  29058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-ne 2481  df-in 3193  df-ss 3200  df-pss 3202
  Copyright terms: Public domain W3C validator