Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afveq12d Structured version   Unicode version

Theorem afveq12d 32008
Description: Equality deduction for function value, analogous to fveq12d 5878. (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 32004 . . 3  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
41, 2fveq12d 5878 . . 3  |-  ( ph  ->  ( F `  A
)  =  ( G `
 B ) )
5 eqidd 2468 . . 3  |-  ( ph  ->  _V  =  _V )
63, 4, 5ifbieq12d 3972 . 2  |-  ( ph  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  if ( G defAt  B ,  ( G `
 B ) ,  _V ) )
7 dfafv2 32007 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
8 dfafv2 32007 . 2  |-  ( G''' B )  =  if ( G defAt  B , 
( G `  B
) ,  _V )
96, 7, 83eqtr4g 2533 1  |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   _Vcvv 3118   ifcif 3945   ` cfv 5594   defAt wdfat 31988  '''cafv 31989
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-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602  df-dfat 31991  df-afv 31992
This theorem is referenced by:  afveq1  32009  afveq2  32010  csbafv12g  32012  afvco2  32051  aoveq123d  32053
  Copyright terms: Public domain W3C validator