Theorem fveqeq2d 5488

Description: Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)

Hypothesis

Ref	Expression
fveqeq2d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
fveqeq2d	|- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )

Proof of Theorem fveqeq2d

Step	Hyp	Ref	Expression
1		fveqeq2d.1	. . 3 |- ( ph -> A = B )
2	1	fveq2d 5484	. 2 |- ( ph -> ( F ` A ) = ( F ` B ) )
3	2	eqeq1d 2173	1 |- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1342   ` cfv 5182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-iota 5147  df-fv 5190

This theorem is referenced by:  fveqeq2  5489  nnnninfeq2  7084  enmkvlem  7116  algcvga  11962  pilem3  13245  nninfomni  13733