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Theorem imbrov2fvoveq 5799
Description: Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
Hypothesis
Ref Expression
imbrov2fvoveq.1  |-  ( X  =  Y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
imbrov2fvoveq  |-  ( X  =  Y  ->  (
( ph  ->  ( F `
 ( ( G `
 X )  .x.  O ) ) R A )  <->  ( ps  ->  ( F `  (
( G `  Y
)  .x.  O )
) R A ) ) )

Proof of Theorem imbrov2fvoveq
StepHypRef Expression
1 imbrov2fvoveq.1 . 2  |-  ( X  =  Y  ->  ( ph 
<->  ps ) )
2 fveq2 5421 . . . 4  |-  ( X  =  Y  ->  ( G `  X )  =  ( G `  Y ) )
32fvoveq1d 5796 . . 3  |-  ( X  =  Y  ->  ( F `  ( ( G `  X )  .x.  O ) )  =  ( F `  (
( G `  Y
)  .x.  O )
) )
43breq1d 3939 . 2  |-  ( X  =  Y  ->  (
( F `  (
( G `  X
)  .x.  O )
) R A  <->  ( F `  ( ( G `  Y )  .x.  O
) ) R A ) )
51, 4imbi12d 233 1  |-  ( X  =  Y  ->  (
( ph  ->  ( F `
 ( ( G `
 X )  .x.  O ) ) R A )  <->  ( ps  ->  ( F `  (
( G `  Y
)  .x.  O )
) R A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   class class class wbr 3929   ` cfv 5123  (class class class)co 5774
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 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  cncfco  12756  mulcncflem  12768  ivthinclemlopn  12792  ivthinclemuopn  12794  limcimolemlt  12811  eflt  12873
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