Theorem imbrov2fvoveq 7381

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 6827	. . . 4 ⊢ (𝑋 = 𝑌 → (𝐺‘𝑋) = (𝐺‘𝑌))
3	2	fvoveq1d 7378	. . 3 ⊢ (𝑋 = 𝑌 → (𝐹‘((𝐺‘𝑋) · 𝑂)) = (𝐹‘((𝐺‘𝑌) · 𝑂)))
4	3	breq1d 5082	. 2 ⊢ (𝑋 = 𝑌 → ((𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴 ↔ (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴))
5	1, 4	imbi12d 345	1 ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359

This theorem is referenced by:  rlim2  15449  rlimclim1  15498  rlimcn1  15541  climcn1  15545  caucvgrlem  15626  cncfco  24892  ftc1lem4  26024  ftc1lem6  26026  itg2gt0cn  38042  ftc1cnnclem  38058  ftc1cnnc  38059  idlimc  46071  limcperiod  46073  limclner  46094  cncfshift  46317  cncfperiod  46322  fperdvper  46362