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Theorem oprabbidv 5902
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 1605 . 2  |-  F/ x ph
2 nfv 1605 . 2  |-  F/ y
ph
3 nfv 1605 . 2  |-  F/ z
ph
4 oprabbidv.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
51, 2, 3, 4oprabbid 5901 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  =  { <. <. x ,  y
>. ,  z >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   {coprab 5859
This theorem is referenced by:  oprabbii  5903  mpt2eq123dva  5909  mpt2eq3dva  5912  resoprab2  5941  erovlem  6754  prismorcsetlem  25912  prismorcset  25914  morcatset1  25915  domcatfun  25925  codcatfun  25934  isrocatset  25957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-oprab 5862
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