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Theorem altopthbg 34599
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 34592 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2 sneqbg 4802 . . 3 (𝐴𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3 sneqbg 4802 . . . 4 (𝐷𝑊 → ({𝐷} = {𝐵} ↔ 𝐷 = 𝐵))
4 eqcom 2740 . . . 4 ({𝐵} = {𝐷} ↔ {𝐷} = {𝐵})
5 eqcom 2740 . . . 4 (𝐵 = 𝐷𝐷 = 𝐵)
63, 4, 53bitr4g 314 . . 3 (𝐷𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
72, 6bi2anan9 638 . 2 ((𝐴𝑉𝐷𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
81, 7bitrid 283 1 ((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {csn 4587  caltop 34587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-sn 4588  df-pr 4590  df-altop 34589
This theorem is referenced by:  altopthb  34601
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