Theorem fvoveq1d 6040

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 6033	. 2 |- ( ph -> ( A O C ) = ( B O C ) )
3	2	fveq2d 5643	1 |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )

Syntax hints:   -> wi 4   = wceq 1397   ` cfv 5326  (class class class)co 6018

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021

This theorem is referenced by:  fvoveq1  6041  imbrov2fvoveq  6043  seqvalcd  10724  pfxfvlsw  11280  swrdswrd  11290  mpomulcn  15309  mulc1cncf  15332  mulcncflem  15350  mulcncf  15351  limccl  15402  ellimc3apf  15403  limcdifap  15405  limcmpted  15406  limcresi  15409  limccoap  15421  dveflem  15469