Theorem feq3dd 6696

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 6694	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶𝐶))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → 𝐹:𝐴⟶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-ss 3896  df-f 6542

This theorem is used by:  vietalem  34211  cofidf2a  50225  veroquadmodzerod  50984  veroquadnolindfd  50985