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Theorem opabbii 4067
Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
Hypothesis
Ref Expression
opabbii.1  |-  ( ph  <->  ps )
Assertion
Ref Expression
opabbii  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }

Proof of Theorem opabbii
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . 2  |-  z  =  z
2 opabbii.1 . . . 4  |-  ( ph  <->  ps )
32a1i 9 . . 3  |-  ( z  =  z  ->  ( ph 
<->  ps ) )
43opabbidv 4066 . 2  |-  ( z  =  z  ->  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps } )
51, 4ax-mp 5 1  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353   {copab 4060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-opab 4062
This theorem is referenced by:  mptv  4097  fconstmpt  4670  xpundi  4679  xpundir  4680  inxp  4757  cnvco  4808  resopab  4947  opabresid  4956  cnvi  5029  cnvun  5030  cnvin  5032  cnvxp  5043  cnvcnv3  5074  coundi  5126  coundir  5127  mptun  5343  fvopab6  5608  cbvoprab1  5941  cbvoprab12  5943  dmoprabss  5951  mpomptx  5960  resoprab  5965  ov6g  6006  dfoprab3s  6185  dfoprab3  6186  dfoprab4  6187  mapsncnv  6689  xpcomco  6820  dmaddpq  7366  dmmulpq  7367  recmulnqg  7378  enq0enq  7418  ltrelxr  8005  ltxr  9759  shftidt2  10822  lmfval  13352  lmbr  13373  cnmptid  13441
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