Theorem copsex2g 5495

Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5497 to reduce axiom usage. (Revised by SN, 1-Sep-2024.)

Hypothesis

Ref	Expression
copsex2g.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
copsex2g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem copsex2g

Step	Hyp	Ref	Expression
1		eqcom 2732	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2		vex 3465	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3465	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opth 5478	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
5	1, 4	bitri 274	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6	5	anbi1i 622	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑))
7	6	2exbii 1843	. 2 ⊢ (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑))
8		id 22	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
9		copsex2g.1	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
10	8, 9	cgsex2g 3508	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜓))
11	7, 10	bitrid 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ⟨cop 4636

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-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637

This theorem is referenced by:  opelopabga  5535  ov6g  7585  ltresr  11165