Theorem copsex4g 5409

Description: An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)

Hypothesis

Ref	Expression
copsex4g.1	⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
copsex4g	⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐶,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝜓,𝑥,𝑦,𝑧,𝑤   𝑥,𝑅,𝑦,𝑧,𝑤   𝑥,𝑆,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem copsex4g

Step	Hyp	Ref	Expression
1		eqcom 2745	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2		vex 3436	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 3436	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	opth 5391	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
5	1, 4	bitri 274	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6		eqcom 2745	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝐶, 𝐷⟩)
7		vex 3436	. . . . . . . 8 ⊢ 𝑧 ∈ V
8		vex 3436	. . . . . . . 8 ⊢ 𝑤 ∈ V
9	7, 8	opth 5391	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷))
10	6, 9	bitri 274	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩ ↔ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷))
11	5, 10	anbi12i 627	. . . . 5 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
12	11	anbi1i 624	. . . 4 ⊢ (((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ 𝜑))
13	12	a1i 11	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ 𝜑)))
14	13	4exbidv 1929	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ 𝜑)))
15		id 22	. . 3 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
16		copsex4g.1	. . 3 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → (𝜑 ↔ 𝜓))
17	15, 16	cgsex4g 3476	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ 𝜑) ↔ 𝜓))
18	14, 17	bitrd 278	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ⟨cop 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568

This theorem is referenced by:  opbrop  5684  ov3  7435