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Theorem oprabbidv 6346
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 1683 . 2  |-  F/ x ph
2 nfv 1683 . 2  |-  F/ y
ph
3 nfv 1683 . 2  |-  F/ z
ph
4 oprabbidv.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
51, 2, 3, 4oprabbid 6345 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  =  { <. <. x ,  y
>. ,  z >.  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   {coprab 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-oprab 6299
This theorem is referenced by:  oprabbii  6347  mpt2eq123dva  6353  mpt2eq3dva  6356  resoprab2  6394  erovlem  7419  joinfval  15505  meetfval  15519  odumeet  15644  odujoin  15646  mppsval  28757
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