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Theorem oprabbii 5792
Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
oprabbii.1  |-  ( ph  <->  ps )
Assertion
Ref Expression
oprabbii  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z
>.  |  ps }
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem oprabbii
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eqid 2115 . 2  |-  w  =  w
2 oprabbii.1 . . . 4  |-  ( ph  <->  ps )
32a1i 9 . . 3  |-  ( w  =  w  ->  ( ph 
<->  ps ) )
43oprabbidv 5791 . 2  |-  ( w  =  w  ->  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z
>.  |  ps } )
51, 4ax-mp 5 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1314   {coprab 5741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-oprab 5744
This theorem is referenced by:  oprab4  5808  mpov  5827  dfxp3  6058  tposmpo  6144  oviec  6501  dfplpq2  7126  dfmpq2  7127  dfmq0qs  7201  dfplq0qs  7202  addsrpr  7517  mulsrpr  7518  addcnsr  7606  mulcnsr  7607  addvalex  7616
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