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Theorem oprabbidv 5703
Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)
Hypothesis
Ref Expression
oprabbidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
oprabbidv  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  =  { <. <. x ,  y
>. ,  z >.  |  ch } )
Distinct variable groups:    x, z, ph    y, z, ph
Allowed substitution hints:    ps( x, y, z)    ch( x, y, z)

Proof of Theorem oprabbidv
StepHypRef Expression
1 nfv 1466 . 2  |-  F/ x ph
2 nfv 1466 . 2  |-  F/ y
ph
3 nfv 1466 . 2  |-  F/ z
ph
4 oprabbidv.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
51, 2, 3, 4oprabbid 5702 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  =  { <. <. x ,  y
>. ,  z >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   {coprab 5653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-oprab 5656
This theorem is referenced by:  oprabbii  5704  mpt2eq123dva  5710  mpt2eq3dva  5713  resoprab2  5742  erovlem  6382
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