Theorem eqvinop 4607

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		opth 4603	. . . . . . . 8 ⊢ (⟨x, y⟩ = ⟨B, C⟩ ↔ (x = B ∧ y = C))
2		ancom 437	. . . . . . . 8 ⊢ ((x = B ∧ y = C) ↔ (y = C ∧ x = B))
3	1, 2	bitri 240	. . . . . . 7 ⊢ (⟨x, y⟩ = ⟨B, C⟩ ↔ (y = C ∧ x = B))
4	3	anbi2i 675	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ (A = ⟨x, y⟩ ∧ (y = C ∧ x = B)))
5		an13 774	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ (y = C ∧ x = B)) ↔ (x = B ∧ (y = C ∧ A = ⟨x, y⟩)))
6	4, 5	bitri 240	. . . . 5 ⊢ ((A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ (x = B ∧ (y = C ∧ A = ⟨x, y⟩)))
7	6	exbii 1582	. . . 4 ⊢ (∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ ∃y(x = B ∧ (y = C ∧ A = ⟨x, y⟩)))
8		19.42v 1905	. . . 4 ⊢ (∃y(x = B ∧ (y = C ∧ A = ⟨x, y⟩)) ↔ (x = B ∧ ∃y(y = C ∧ A = ⟨x, y⟩)))
9		eqvinop.2	. . . . . 6 ⊢ C ∈ V
10		opeq2 4580	. . . . . . 7 ⊢ (y = C → ⟨x, y⟩ = ⟨x, C⟩)
11	10	eqeq2d 2364	. . . . . 6 ⊢ (y = C → (A = ⟨x, y⟩ ↔ A = ⟨x, C⟩))
12	9, 11	ceqsexv 2895	. . . . 5 ⊢ (∃y(y = C ∧ A = ⟨x, y⟩) ↔ A = ⟨x, C⟩)
13	12	anbi2i 675	. . . 4 ⊢ ((x = B ∧ ∃y(y = C ∧ A = ⟨x, y⟩)) ↔ (x = B ∧ A = ⟨x, C⟩))
14	7, 8, 13	3bitri 262	. . 3 ⊢ (∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ (x = B ∧ A = ⟨x, C⟩))
15	14	exbii 1582	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩) ↔ ∃x(x = B ∧ A = ⟨x, C⟩))
16		eqvinop.1	. . 3 ⊢ B ∈ V
17		opeq1 4579	. . . 4 ⊢ (x = B → ⟨x, C⟩ = ⟨B, C⟩)
18	17	eqeq2d 2364	. . 3 ⊢ (x = B → (A = ⟨x, C⟩ ↔ A = ⟨B, C⟩))
19	16, 18	ceqsexv 2895	. 2 ⊢ (∃x(x = B ∧ A = ⟨x, C⟩) ↔ A = ⟨B, C⟩)
20	15, 19	bitr2i 241	1 ⊢ (A = ⟨B, C⟩ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ = ⟨B, C⟩))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is referenced by:  copsexg  4608  ralxpf  4828  oprabid  5551