Theorem mpteq12daOLD 44677

Description: Obsolete version of mpteq12da 5229 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 2201	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶)
4		mpteq12daOLD.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
5	1, 4	ralrimia 3246	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
6		mpteq12f 5232	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
7	3, 5, 6	syl2anc 582	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1531   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  ∀wral 3051   ↦ cmpt 5227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-opab 5207  df-mpt 5228

This theorem is referenced by: (None)