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Theorem fvoveq1d 5872
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
StepHypRef Expression
1 fvoveq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21oveq1d 5865 . 2  |-  ( ph  ->  ( A O C )  =  ( B O C ) )
32fveq2d 5498 1  |-  ( ph  ->  ( F `  ( A O C ) )  =  ( F `  ( B O C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   ` cfv 5196  (class class class)co 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204  df-ov 5853
This theorem is referenced by:  fvoveq1  5873  imbrov2fvoveq  5875  seqvalcd  10402  mulc1cncf  13329  mulcncflem  13343  mulcncf  13344  limccl  13381  ellimc3apf  13382  limcdifap  13384  limcmpted  13385  limcresi  13388  limccoap  13400  dveflem  13440
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