Theorem mpteq2dfa 46418

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 5199	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2146   ↦ cmpt 5188

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-opab 5170  df-mpt 5189

This theorem is used by: (None)