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 30058
Description: Equality deduction for function value, analogous to fveq12d 5712. (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 30054 . . 3  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
41, 2fveq12d 5712 . . 3  |-  ( ph  ->  ( F `  A
)  =  ( G `
 B ) )
5 eqidd 2444 . . 3  |-  ( ph  ->  _V  =  _V )
63, 4, 5ifbieq12d 3831 . 2  |-  ( ph  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  if ( G defAt  B ,  ( G `
 B ) ,  _V ) )
7 dfafv2 30057 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
8 dfafv2 30057 . 2  |-  ( G''' B )  =  if ( G defAt  B , 
( G `  B
) ,  _V )
96, 7, 83eqtr4g 2500 1  |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   _Vcvv 2987   ifcif 3806   ` cfv 5433   defAt wdfat 30036  '''cafv 30037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2736  df-rab 2739  df-v 2989  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-res 4867  df-iota 5396  df-fun 5435  df-fv 5441  df-dfat 30039  df-afv 30040
This theorem is referenced by:  afveq1  30059  afveq2  30060  csbafv12g  30062  afvco2  30101  aoveq123d  30103
  Copyright terms: Public domain W3C validator