Theorem fveqeq2d 5524

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 5520	. 2 |- ( ph -> ( F ` A ) = ( F ` B ) )
3	2	eqeq1d 2186	1 |- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   ` cfv 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225

This theorem is referenced by:  fveqeq2  5525  nnnninfeq2  7127  enmkvlem  7159  algcvga  12051  pilem3  14207  nninfomni  14771