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Theorem opthg2 5378
Description: Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opthg2 ((𝐶𝑉𝐷𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem opthg2
StepHypRef Expression
1 opthg 5376 . 2 ((𝐶𝑉𝐷𝑊) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
2 eqcom 2745 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
3 eqcom 2745 . . 3 (𝐴 = 𝐶𝐶 = 𝐴)
4 eqcom 2745 . . 3 (𝐵 = 𝐷𝐷 = 𝐵)
53, 4anbi12i 630 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐶 = 𝐴𝐷 = 𝐵))
61, 2, 53bitr4g 317 1 ((𝐶𝑉𝐷𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  cop 4562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563
This theorem is referenced by:  opth2  5379  brsnop  5420  fliftel  7137  symg2bas  18810  projf1o  42438
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