Theorem eqvinop 1901

Description: A variable introduction law for ordered pairs. Analogue of Lemma 15 of [Monk2] p. 109.

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 group(s):   x,y,A   x,B,y   x,C,y

Proof of Theorem eqvinop

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 ⊢ x ∈ V
2		visset 1350	. . . . . . . 8 ⊢ y ∈ V
3		eqvinop.2	. . . . . . . 8 ⊢ C ∈ V
4	1, 2, 3	opth 1898	. . . . . . 7 ⊢ (⟨x, y⟩ = ⟨B, C⟩ ↔ (x = B ∧ y = C))
5	4	anbi2i 367	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ (A = ⟨x, y⟩ ∧ (x = B ∧ y = C)))
6		ancom 333	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ (x = B ∧ y = C)) ↔ ((x = B ∧ y = C) ∧ A = ⟨x, y⟩))
7		anass 336	. . . . . 6 ⊢ (((x = B ∧ y = C) ∧ A = ⟨x, y⟩) ↔ (x = B ∧ (y = C ∧ A = ⟨x, y⟩)))
8	5, 6, 7	3bitr 155	. . . . 5 ⊢ ((A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ (x = B ∧ (y = C ∧ A = ⟨x, y⟩)))
9	8	biex 733	. . . 4 ⊢ (∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ ∃y(x = B ∧ (y = C ∧ A = ⟨x, y⟩)))
10		19.42v 966	. . . 4 ⊢ (∃y(x = B ∧ (y = C ∧ A = ⟨x, y⟩)) ↔ (x = B ∧ ∃y(y = C ∧ A = ⟨x, y⟩)))
11		opeq2 1877	. . . . . . 7 ⊢ (y = C → ⟨x, y⟩ = ⟨x, C⟩)
12	11	cleq2d 1112	. . . . . 6 ⊢ (y = C → (A = ⟨x, y⟩ ↔ A = ⟨x, C⟩))
13	3, 12	ceqsexv 1371	. . . . 5 ⊢ (∃y(y = C ∧ A = ⟨x, y⟩) ↔ A = ⟨x, C⟩)
14	13	anbi2i 367	. . . 4 ⊢ ((x = B ∧ ∃y(y = C ∧ A = ⟨x, y⟩)) ↔ (x = B ∧ A = ⟨x, C⟩))
15	9, 10, 14	3bitr 155	. . 3 ⊢ (∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ (x = B ∧ A = ⟨x, C⟩))
16	15	biex 733	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ ∃x(x = B ∧ A = ⟨x, C⟩))
17		eqvinop.1	. . 3 ⊢ B ∈ V
18		opeq1 1876	. . . 4 ⊢ (x = B → ⟨x, C⟩ = ⟨B, C⟩)
19	18	cleq2d 1112	. . 3 ⊢ (x = B → (A = ⟨x, C⟩ ↔ A = ⟨B, C⟩))
20	17, 19	ceqsexv 1371	. 2 ⊢ (∃x(x = B ∧ A = ⟨x, C⟩) ↔ A = ⟨B, C⟩)
21	16, 20	bitr2 152	1 ⊢ (A = ⟨B, C⟩ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810

This theorem is referenced by:  copsexg 1902

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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