Theorem feq2dd 6714

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 6712	. 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 1570  ⟶wf 6551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-fn 6558  df-f 6559

This theorem is used by:  elcgrabasi  29402  elrspunidl  34045  1arithidomlem2  34135  1arithidom  34136  selvply1rhmlemb  34218  selvply1rhm0  34225  evlextv  34241  vieta  34279  evlselvlem  43807  cncfuni  47099  oppfdiag1  50752  oppfdiag  50754  wrdf1d  51218