Theorem eqvinop 3971

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 3968	. . . . . . 7 ⊢ (⟨x, y⟩ = ⟨B, 𝐶⟩ ↔ (x = B ∧ y = 𝐶))
4	3	anbi2i 430	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, 𝐶⟩) ↔ (A = ⟨x, y⟩ ∧ (x = B ∧ y = 𝐶)))
5		ancom 253	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ (x = B ∧ y = 𝐶)) ↔ ((x = B ∧ y = 𝐶) ∧ A = ⟨x, y⟩))
6		anass 381	. . . . . 6 ⊢ (((x = B ∧ y = 𝐶) ∧ A = ⟨x, y⟩) ↔ (x = B ∧ (y = 𝐶 ∧ A = ⟨x, y⟩)))
7	4, 5, 6	3bitri 195	. . . . 5 ⊢ ((A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, 𝐶⟩) ↔ (x = B ∧ (y = 𝐶 ∧ A = ⟨x, y⟩)))
8	7	exbii 1493	. . . 4 ⊢ (∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, 𝐶⟩) ↔ ∃y(x = B ∧ (y = 𝐶 ∧ A = ⟨x, y⟩)))
9		19.42v 1783	. . . 4 ⊢ (∃y(x = B ∧ (y = 𝐶 ∧ A = ⟨x, y⟩)) ↔ (x = B ∧ ∃y(y = 𝐶 ∧ A = ⟨x, y⟩)))
10		opeq2 3541	. . . . . . 7 ⊢ (y = 𝐶 → ⟨x, y⟩ = ⟨x, 𝐶⟩)
11	10	eqeq2d 2048	. . . . . 6 ⊢ (y = 𝐶 → (A = ⟨x, y⟩ ↔ A = ⟨x, 𝐶⟩))
12	2, 11	ceqsexv 2587	. . . . 5 ⊢ (∃y(y = 𝐶 ∧ A = ⟨x, y⟩) ↔ A = ⟨x, 𝐶⟩)
13	12	anbi2i 430	. . . 4 ⊢ ((x = B ∧ ∃y(y = 𝐶 ∧ A = ⟨x, y⟩)) ↔ (x = B ∧ A = ⟨x, 𝐶⟩))
14	8, 9, 13	3bitri 195	. . 3 ⊢ (∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, 𝐶⟩) ↔ (x = B ∧ A = ⟨x, 𝐶⟩))
15	14	exbii 1493	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, 𝐶⟩) ↔ ∃x(x = B ∧ A = ⟨x, 𝐶⟩))
16		opeq1 3540	. . . 4 ⊢ (x = B → ⟨x, 𝐶⟩ = ⟨B, 𝐶⟩)
17	16	eqeq2d 2048	. . 3 ⊢ (x = B → (A = ⟨x, 𝐶⟩ ↔ A = ⟨B, 𝐶⟩))
18	1, 17	ceqsexv 2587	. 2 ⊢ (∃x(x = B ∧ A = ⟨x, 𝐶⟩) ↔ A = ⟨B, 𝐶⟩)
19	15, 18	bitr2i 174	1 ⊢ (A = ⟨B, 𝐶⟩ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, 𝐶⟩))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376

This theorem is referenced by:  copsexg  3972  ralxpf  4425  rexxpf  4426  oprabid  5480