Theorem feq3dd 32560

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

Hypotheses

Ref	Expression
feq3dd.eq	⊢ (𝜑 → 𝐵 = 𝐶)
feq3dd.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
feq3dd	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Proof of Theorem feq3dd

Step	Hyp	Ref	Expression
1		feq3dd.f	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		feq3dd.eq	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
3	2	feq3d 6702	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶𝐶))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   = wceq 1539  ⟶wf 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2726  df-ss 3948  df-f 6544

This theorem is referenced by: (None)