Theorem mpteq12daOLD 42649

Description: Obsolete version of mpteq12da 5154 as of 11-Nov-2024. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mpteq12daOLD.1	⊢ Ⅎ𝑥𝜑
mpteq12daOLD.2	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12daOLD.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12daOLD	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Proof of Theorem mpteq12daOLD

Step	Hyp	Ref	Expression
1		mpteq12daOLD.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		mpteq12daOLD.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
3	1, 2	alrimi 2213	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶)
4		mpteq12daOLD.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
5	1, 4	ralrimia 3421	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
6		mpteq12f 5157	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
7	3, 5, 6	syl2anc 587	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1541   = wceq 1543  Ⅎwnf 1791   ∈ wcel 2112  ∀wral 3064   ↦ cmpt 5152

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 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2177  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-opab 5133  df-mpt 5153

This theorem is referenced by: (None)