Theorem opelopab4 44576

Description: Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5531. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
opelopab4	⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem opelopab4

Step	Hyp	Ref	Expression
1		elopab 5531	. 2 ⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2		vex 3483	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3483	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opth 5480	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
5		eqcom 2743	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
6	4, 5	bitr3i 277	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
7	6	anbi1i 624	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8	7	2exbii 1848	. 2 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
9	1, 8	bitr4i 278	1 ⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ⟨cop 4631  {copab 5204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205

This theorem is referenced by: (None)