Theorem imbrov2fvoveq 5947

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 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
imbrov2fvoveq	⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)))

Proof of Theorem imbrov2fvoveq

Step	Hyp	Ref	Expression
1		imbrov2fvoveq.1	. 2 ⊢ (𝑋 = 𝑌 → (𝜑 ↔ 𝜓))
2		fveq2 5558	. . . 4 ⊢ (𝑋 = 𝑌 → (𝐺‘𝑋) = (𝐺‘𝑌))
3	2	fvoveq1d 5944	. . 3 ⊢ (𝑋 = 𝑌 → (𝐹‘((𝐺‘𝑋) · 𝑂)) = (𝐹‘((𝐺‘𝑌) · 𝑂)))
4	3	breq1d 4043	. 2 ⊢ (𝑋 = 𝑌 → ((𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴 ↔ (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴))
5	1, 4	imbi12d 234	1 ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   class class class wbr 4033  ‘cfv 5258  (class class class)co 5922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  cncfco  14827  mulcncflem  14843  ivthinclemlopn  14872  ivthinclemuopn  14874  limcimolemlt  14900  eflt  15011