Theorem fvoveq1d 6029

Description: Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)

Hypothesis

Ref	Expression
fvoveq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
fvoveq1d	|- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )

Proof of Theorem fvoveq1d

Step	Hyp	Ref	Expression
1		fvoveq1d.1	. . 3 |- ( ph -> A = B )
2	1	oveq1d 6022	. 2 |- ( ph -> ( A O C ) = ( B O C ) )
3	2	fveq2d 5633	1 |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )

Syntax hints:   -> wi 4   = wceq 1395   ` cfv 5318  (class class class)co 6007

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6010

This theorem is referenced by:  fvoveq1  6030  imbrov2fvoveq  6032  seqvalcd  10695  pfxfvlsw  11243  swrdswrd  11253  mpomulcn  15256  mulc1cncf  15279  mulcncflem  15297  mulcncf  15298  limccl  15349  ellimc3apf  15350  limcdifap  15352  limcmpted  15353  limcresi  15356  limccoap  15368  dveflem  15416