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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 → (𝜑𝜓))
Assertion
Ref Expression
imbrov2fvoveq (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺𝑌) · 𝑂))𝑅𝐴)))

Proof of Theorem imbrov2fvoveq
StepHypRef Expression
1 imbrov2fvoveq.1 . 2 (𝑋 = 𝑌 → (𝜑𝜓))
2 fveq2 5421 . . . 4 (𝑋 = 𝑌 → (𝐺𝑋) = (𝐺𝑌))
32fvoveq1d 5796 . . 3 (𝑋 = 𝑌 → (𝐹‘((𝐺𝑋) · 𝑂)) = (𝐹‘((𝐺𝑌) · 𝑂)))
43breq1d 3939 . 2 (𝑋 = 𝑌 → ((𝐹‘((𝐺𝑋) · 𝑂))𝑅𝐴 ↔ (𝐹‘((𝐺𝑌) · 𝑂))𝑅𝐴))
51, 4imbi12d 233 1 (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺𝑌) · 𝑂))𝑅𝐴)))
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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