Theorem feq3dd 6680

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 6678	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶𝐶))
4	1, 3	mpbid 234	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   = wceq 1562  ⟶wf 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-cleq 2756  df-ss 3923  df-f 6527

This theorem is referenced by:  vietalem  33878  cofidf2a  49743