Theorem fvoveq1d 6072

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 6065	. 2 |- ( ph -> ( A O C ) = ( B O C ) )
3	2	fveq2d 5674	1 |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )

Syntax hints:   -> wi 4   = wceq 1398   ` cfv 5352  (class class class)co 6050

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  fvoveq1  6073  imbrov2fvoveq  6075  seqvalcd  10823  pfxfvlsw  11387  swrdswrd  11397  mpomulcn  15431  mulc1cncf  15454  mulcncflem  15472  mulcncf  15473  limccl  15524  ellimc3apf  15525  limcdifap  15527  limcmpted  15528  limcresi  15531  limccoap  15543  dveflem  15591