Theorem mpteq12daOLD 45151

Description: Obsolete version of mpteq12da 5251 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 2214	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶)
4		mpteq12daOLD.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
5	1, 4	ralrimia 3264	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
6		mpteq12f 5254	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
7	3, 5, 6	syl2anc 583	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  ∀wral 3067   ↦ cmpt 5249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-opab 5229  df-mpt 5250

This theorem is referenced by: (None)