Theorem fveqeq2d 5494

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 5490	. 2 |- ( ph -> ( F ` A ) = ( F ` B ) )
3	2	eqeq1d 2174	1 |- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   ` cfv 5188

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196

This theorem is referenced by:  fveqeq2  5495  nnnninfeq2  7093  enmkvlem  7125  algcvga  11983  pilem3  13344  nninfomni  13899