Theorem fveqeq2d 5703

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 5699	. 2 |- ( ph -> ( F ` A ) = ( F ` B ) )
3	2	eqeq1d 2247	1 |- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385

This theorem is used by:  fveqeq2  5704  nnnninfeq2  7469  enmkvlem  7501  nninfctlemfo  12817  algcvga  12829  mhmex  13769  resmhm  13794  isghm  14046  lspsneq0  14763  pilem3  15884  2lgslem3c  16214  2lgslem3d  16215  nninfomni  17062  qdiff  17098