Theorem fveq12d 5421

Description: Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)

Hypotheses

Ref	Expression
fveq12d.1	|- ( ph -> F = G )
fveq12d.2	|- ( ph -> A = B )

Assertion

Ref	Expression
fveq12d	|- ( ph -> ( F ` A ) = ( G ` B ) )

Proof of Theorem fveq12d

Step	Hyp	Ref	Expression
1		fveq12d.1	. . 3 |- ( ph -> F = G )
2	1	fveq1d 5416	. 2 |- ( ph -> ( F ` A ) = ( G ` A ) )
3		fveq12d.2	. . 3 |- ( ph -> A = B )
4	3	fveq2d 5418	. 2 |- ( ph -> ( G ` A ) = ( G ` B ) )
5	2, 4	eqtrd 2170	1 |- ( ph -> ( F ` A ) = ( G ` B ) )

Syntax hints:   -> wi 4   = wceq 1331   ` cfv 5118

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 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126

This theorem is referenced by:  nffvd  5426  fvsng  5609  tfrlem3ag  6199  tfrlem3a  6200  tfrlemi1  6222  tfr1onlem3ag  6227  omp1eomlem  6972  seq3shft  10603  climshft2  11068  fsum3  11149  ctiunctlemfo  11941  reldvg  12806  dvfvalap  12808