Theorem fvoveq1d 6050

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 6043	. 2 |- ( ph -> ( A O C ) = ( B O C ) )
3	2	fveq2d 5652	1 |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )

Syntax hints:   -> wi 4   = wceq 1398   ` cfv 5333  (class class class)co 6028

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 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  fvoveq1  6051  imbrov2fvoveq  6053  seqvalcd  10786  pfxfvlsw  11342  swrdswrd  11352  mpomulcn  15377  mulc1cncf  15400  mulcncflem  15418  mulcncf  15419  limccl  15470  ellimc3apf  15471  limcdifap  15473  limcmpted  15474  limcresi  15477  limccoap  15489  dveflem  15537