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Theorem afveq12d 27975
Description: Equality deduction for function value, analogous to fveq12d 5736. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
afveq12d.1  |-  ( ph  ->  F  =  G )
afveq12d.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
afveq12d  |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )

Proof of Theorem afveq12d
StepHypRef Expression
1 afveq12d.1 . . . 4  |-  ( ph  ->  F  =  G )
2 afveq12d.2 . . . 4  |-  ( ph  ->  A  =  B )
31, 2dfateq12d 27971 . . 3  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
41, 2fveq12d 5736 . . 3  |-  ( ph  ->  ( F `  A
)  =  ( G `
 B ) )
5 eqidd 2439 . . 3  |-  ( ph  ->  _V  =  _V )
63, 4, 5ifbieq12d 3763 . 2  |-  ( ph  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  if ( G defAt  B ,  ( G `
 B ) ,  _V ) )
7 dfafv2 27974 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
8 dfafv2 27974 . 2  |-  ( G''' B )  =  if ( G defAt  B , 
( G `  B
) ,  _V )
96, 7, 83eqtr4g 2495 1  |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   _Vcvv 2958   ifcif 3741   ` cfv 5456   defAt wdfat 27949  '''cafv 27950
This theorem is referenced by:  afveq1  27976  afveq2  27977  csbafv12g  27979  afvco2  28018  aoveq123d  28020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fv 5464  df-dfat 27952  df-afv 27953
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