Theorem fveqeq2d 5429

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 5425	. 2 |- ( ph -> ( F ` A ) = ( F ` B ) )
3	2	eqeq1d 2148	1 |- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   ` cfv 5123

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131

This theorem is referenced by:  fveqeq2  5430  algcvga  11732  pilem3  12864