Theorem fvoveq1d 6023

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 6016	. 2 |- ( ph -> ( A O C ) = ( B O C ) )
3	2	fveq2d 5631	1 |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )

Syntax hints:   -> wi 4   = wceq 1395   ` cfv 5318  (class class class)co 6001

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 6004

This theorem is referenced by:  fvoveq1  6024  imbrov2fvoveq  6026  seqvalcd  10683  pfxfvlsw  11227  swrdswrd  11237  mpomulcn  15240  mulc1cncf  15263  mulcncflem  15281  mulcncf  15282  limccl  15333  ellimc3apf  15334  limcdifap  15336  limcmpted  15337  limcresi  15340  limccoap  15352  dveflem  15400