Theorem feq2dd 6691

Description: Equality deduction for functions. (Contributed by Thierry Arnoux, 27-May-2025.)

Hypotheses

Ref	Expression
feq2dd.eq	⊢ (𝜑 → 𝐴 = 𝐵)
feq2dd.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Assertion

Ref	Expression
feq2dd	⊢ (𝜑 → 𝐹:𝐵⟶𝐶)

Proof of Theorem feq2dd

Step	Hyp	Ref	Expression
1		feq2dd.f	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
2		feq2dd.eq	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	feq2d 6689	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ⟶wf 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-fn 6539  df-f 6540

This theorem is used by:  1arithidomlem2  33835  1arithidom  33836  selvply1rhmlemb  33918  selvply1rhm0  33925  evlextv  33941  vieta  33979  evlselvlem  43348  cncfuni  46628  oppfdiag1  50220  oppfdiag  50222