Theorem mpteq12dfOLD 5157

Description: Obsolete version of mpteq12df 5156 as of 11-Nov-2024. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mpteq12df.1	⊢ Ⅎ𝑥𝜑
mpteq12df.2	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12df.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12dfOLD	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Proof of Theorem mpteq12dfOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq12df.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		nfv 1918	. . 3 ⊢ Ⅎ𝑦𝜑
3		mpteq12df.2	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eleq2d 2824	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
5		mpteq12df.3	. . . . 5 ⊢ (𝜑 → 𝐵 = 𝐷)
6	5	eqeq2d 2749	. . . 4 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
7	4, 6	anbi12d 630	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
8	1, 2, 7	opabbid 5135	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
9		df-mpt 5154	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
10		df-mpt 5154	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
11	8, 9, 10	3eqtr4g 2804	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  {copab 5132   ↦ cmpt 5153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-opab 5133  df-mpt 5154

This theorem is referenced by: (None)