Theorem feq2dd 6695

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 6693	. 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 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-fn 6541  df-f 6542

This theorem is used by:  elcgrabasi  29375  1arithidomlem2  34068  1arithidom  34069  selvply1rhmlemb  34151  selvply1rhm0  34158  evlextv  34174  vieta  34212  evlselvlem  43617  cncfuni  46896  oppfdiag1  50522  oppfdiag  50524  wrdf1d  50940