Theorem afveq12d 41213

Description: Equality deduction for function value, analogous to fveq12d 6197. (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

Step	Hyp	Ref	Expression
1		afveq12d.1	. . . 4 |- ( ph -> F = G )
2		afveq12d.2	. . . 4 |- ( ph -> A = B )
3	1, 2	dfateq12d 41209	. . 3 |- ( ph -> ( F defAt A <-> G defAt B ) )
4	1, 2	fveq12d 6197	. . 3 |- ( ph -> ( F ` A ) = ( G ` B ) )
5	3, 4	ifbieq1d 4109	. 2 |- ( ph -> if ( F defAt A , ( F ` A ) , _V ) = if ( G defAt B , ( G ` B ) , _V ) )
6		dfafv2 41212	. 2 |- ( F''' A ) = if ( F defAt A , ( F ` A ) , _V )
7		dfafv2 41212	. 2 |- ( G''' B ) = if ( G defAt B , ( G ` B ) , _V )
8	5, 6, 7	3eqtr4g 2681	1 |- ( ph -> ( F''' A ) = ( G''' B ) )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   ifcif 4086   ` cfv 5888   defAt wdfat 41193  '''cafv 41194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  afveq1  41214  afveq2  41215  csbafv12g  41217  afvco2  41256  aoveq123d  41258