Theorem opthg2 5459

Description: Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opthg2	⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem opthg2

Step	Hyp	Ref	Expression
1		opthg 5457	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
2		eqcom 2743	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
3		eqcom 2743	. . 3 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
4		eqcom 2743	. . 3 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵)
5	3, 4	anbi12i 628	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
6	1, 2, 5	3bitr4g 314	1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613

This theorem is referenced by:  opth2  5460  brsnop  5502  fliftel  7307  symg2bas  19379  projf1o  45188