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Theorem afveq12d 27996
Description: Equality deduction for function value, analogous to fveq12d 5531. (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 27992 . . 3  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
41, 2fveq12d 5531 . . 3  |-  ( ph  ->  ( F `  A
)  =  ( G `
 B ) )
5 eqidd 2284 . . 3  |-  ( ph  ->  _V  =  _V )
63, 4, 5ifbieq12d 3587 . 2  |-  ( ph  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  if ( G defAt  B ,  ( G `
 B ) ,  _V ) )
7 dfafv2 27995 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
8 dfafv2 27995 . 2  |-  ( G''' B )  =  if ( G defAt  B , 
( G `  B
) ,  _V )
96, 7, 83eqtr4g 2340 1  |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   _Vcvv 2788   ifcif 3565   ` cfv 5255   defAt wdfat 27971  '''cafv 27972
This theorem is referenced by:  afveq1  27997  afveq2  27998  csbafv12g  28000  afvco2  28037  aoveq123d  28038  ffnaov  28059
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-dfat 27974  df-afv 27975
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