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Theorem altopthbg 33486
 Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
Assertion
Ref Expression
altopthbg ((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem altopthbg
StepHypRef Expression
1 altopthsn 33479 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2 sneqbg 4758 . . 3 (𝐴𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3 sneqbg 4758 . . . 4 (𝐷𝑊 → ({𝐷} = {𝐵} ↔ 𝐷 = 𝐵))
4 eqcom 2831 . . . 4 ({𝐵} = {𝐷} ↔ {𝐷} = {𝐵})
5 eqcom 2831 . . . 4 (𝐵 = 𝐷𝐷 = 𝐵)
63, 4, 53bitr4g 317 . . 3 (𝐷𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
72, 6bi2anan9 638 . 2 ((𝐴𝑉𝐷𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
81, 7syl5bb 286 1 ((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {csn 4550  ⟪caltop 33474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-altop 33476 This theorem is referenced by:  altopthb  33488
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