Theorem mpteq2dfa 45725

Description: Slightly more general equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 21-Dec-2024.)

Hypotheses

Ref	Expression
mpteq2dfa.1	⊢ Ⅎ𝑥𝜑
mpteq2dfa.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
mpteq2dfa	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

Proof of Theorem mpteq2dfa

Step	Hyp	Ref	Expression
1		mpteq2dfa.1	. 2 ⊢ Ⅎ𝑥𝜑
2		mpteq2dfa.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	1, 2	mpteq2da 5167	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121   ↦ cmpt 5156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-opab 5138  df-mpt 5157

This theorem is referenced by: (None)