Theorem eqvinop 5425

Description: A variable introduction law for ordered pairs. Analogue of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)

Hypotheses

Ref	Expression
eqvinop.1	⊢ 𝐵 ∈ V
eqvinop.2	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
eqvinop	⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))

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

Proof of Theorem eqvinop

Step	Hyp	Ref	Expression
1		eqvinop.1	. . . . . . . 8 ⊢ 𝐵 ∈ V
2		eqvinop.2	. . . . . . . 8 ⊢ 𝐶 ∈ V
3	1, 2	opth2 5418	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩ ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶))
4	3	anbi2i 623	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)))
5		ancom 460	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) ↔ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
6		anass 468	. . . . . 6 ⊢ (((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
7	4, 5, 6	3bitri 297	. . . . 5 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
8	7	exbii 1849	. . . 4 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
9		19.42v 1954	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
10		opeq2 4823	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐶⟩)
11	10	eqeq2d 2742	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝐶⟩))
12	2, 11	ceqsexv 3486	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ 𝐴 = ⟨𝑥, 𝐶⟩)
13	12	anbi2i 623	. . . 4 ⊢ ((𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶⟩))
14	8, 9, 13	3bitri 297	. . 3 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶⟩))
15	14	exbii 1849	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶⟩))
16		opeq1 4822	. . . 4 ⊢ (𝑥 = 𝐵 → ⟨𝑥, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
17	16	eqeq2d 2742	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 = ⟨𝑥, 𝐶⟩ ↔ 𝐴 = ⟨𝐵, 𝐶⟩))
18	1, 17	ceqsexv 3486	. 2 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶⟩) ↔ 𝐴 = ⟨𝐵, 𝐶⟩)
19	15, 18	bitr2i 276	1 ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  ⟨cop 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580

This theorem is referenced by:  copsexgw  5428  copsexg  5429  ralxpf  5785  oprabidw  7377  oprabid  7378