Theorem feq2dd 32654

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 6730	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)

Syntax hints:   → wi 4   = wceq 1539  ⟶wf 6565

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 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2729  df-fn 6572  df-f 6573

This theorem is referenced by:  1arithidomlem2  33576  1arithidom  33577