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Theorem oprabbidv 6128
Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (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 1629 . 2  |-  F/ x ph
2 nfv 1629 . 2  |-  F/ y
ph
3 nfv 1629 . 2  |-  F/ z
ph
4 oprabbidv.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
51, 2, 3, 4oprabbid 6127 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  =  { <. <. x ,  y
>. ,  z >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   {coprab 6082
This theorem is referenced by:  oprabbii  6129  mpt2eq123dva  6135  mpt2eq3dva  6138  resoprab2  6167  erovlem  7000
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 2417
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 2423  df-cleq 2429  df-oprab 6085
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