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Theorem fvoveq1d 5799
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 5792 . 2  |-  ( ph  ->  ( A O C )  =  ( B O C ) )
32fveq2d 5428 1  |-  ( ph  ->  ( F `  ( A O C ) )  =  ( F `  ( B O C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   ` cfv 5126  (class class class)co 5777
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-iota 5091  df-fv 5134  df-ov 5780
This theorem is referenced by:  fvoveq1  5800  imbrov2fvoveq  5802  seqvalcd  10256  mulc1cncf  12771  mulcncflem  12785  mulcncf  12786  limccl  12823  ellimc3apf  12824  limcdifap  12826  limcmpted  12827  limcresi  12830  limccoap  12842  dveflem  12882
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