Theorem opthg2 5447

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 5445	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
2		eqcom 2769	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
3		eqcom 2769	. . 3 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
4		eqcom 2769	. . 3 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵)
5	3, 4	anbi12i 637	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
6	1, 2, 5	3bitr4g 316	1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  opth2  5448  brsnop  5492  fliftel  7293  symg2bas  19433  projf1o  45774  gpgedgiov  48687