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Theorem imbrov2fvoveq 5669
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 5299 . . . 4  |-  ( X  =  Y  ->  ( G `  X )  =  ( G `  Y ) )
32fvoveq1d 5666 . . 3  |-  ( X  =  Y  ->  ( F `  ( ( G `  X )  .x.  O ) )  =  ( F `  (
( G `  Y
)  .x.  O )
) )
43breq1d 3853 . 2  |-  ( X  =  Y  ->  (
( F `  (
( G `  X
)  .x.  O )
) R A  <->  ( F `  ( ( G `  Y )  .x.  O
) ) R A ) )
51, 4imbi12d 232 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 103    = wceq 1289   class class class wbr 3843   ` cfv 5010  (class class class)co 5644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-iota 4975  df-fv 5018  df-ov 5647
This theorem is referenced by:  cncfco  11530
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