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Theorem imbrov2fvoveq 5875
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 5494 . . . 4 (𝑋 = 𝑌 → (𝐺𝑋) = (𝐺𝑌))
32fvoveq1d 5872 . . 3 (𝑋 = 𝑌 → (𝐹‘((𝐺𝑋) · 𝑂)) = (𝐹‘((𝐺𝑌) · 𝑂)))
43breq1d 3997 . 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 1348   class class class wbr 3987  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:  cncfco  13293  mulcncflem  13305  ivthinclemlopn  13329  ivthinclemuopn  13331  limcimolemlt  13348  eflt  13411
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