Theorem imbrov2fvoveq 7435

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 6881	. . . 4 ⊢ (𝑋 = 𝑌 → (𝐺‘𝑋) = (𝐺‘𝑌))
3	2	fvoveq1d 7432	. . 3 ⊢ (𝑋 = 𝑌 → (𝐹‘((𝐺‘𝑋) · 𝑂)) = (𝐹‘((𝐺‘𝑌) · 𝑂)))
4	3	breq1d 5119	. 2 ⊢ (𝑋 = 𝑌 → ((𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴 ↔ (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴))
5	1, 4	imbi12d 347	1 ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by:  rlim2  15543  rlimclim1  15592  rlimcn1  15635  climcn1  15639  caucvgrlem  15720  cncfco  25066  ftc1lem4  26198  ftc1lem6  26200  itg2gt0cn  38346  ftc1cnnclem  38362  ftc1cnnc  38363  idlimc  46362  limcperiod  46364  limclner  46385  cncfshift  46608  cncfperiod  46613  fperdvper  46653