Theorem eqvinop 4286

Description: A variable introduction law for ordered pairs. Analog 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 4283	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩ ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶))
4	3	anbi2i 457	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)))
5		ancom 266	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) ↔ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
6		anass 401	. . . . . 6 ⊢ (((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
7	4, 5, 6	3bitri 206	. . . . 5 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
8	7	exbii 1627	. . . 4 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
9		19.42v 1929	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
10		opeq2 3819	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐶⟩)
11	10	eqeq2d 2216	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝐶⟩))
12	2, 11	ceqsexv 2810	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ 𝐴 = ⟨𝑥, 𝐶⟩)
13	12	anbi2i 457	. . . 4 ⊢ ((𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶⟩))
14	8, 9, 13	3bitri 206	. . 3 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶⟩))
15	14	exbii 1627	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶⟩))
16		opeq1 3818	. . . 4 ⊢ (𝑥 = 𝐵 → ⟨𝑥, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
17	16	eqeq2d 2216	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 = ⟨𝑥, 𝐶⟩ ↔ 𝐴 = ⟨𝐵, 𝐶⟩))
18	1, 17	ceqsexv 2810	. 2 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶⟩) ↔ 𝐴 = ⟨𝐵, 𝐶⟩)
19	15, 18	bitr2i 185	1 ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1372  ∃wex 1514   ∈ wcel 2175  Vcvv 2771  ⟨cop 3635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641

This theorem is referenced by:  copsexg  4287  ralxpf  4823  rexxpf  4824  oprabid  5975