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Theorem altopthg 35472
Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopthg ((𝐴𝑉𝐵𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem altopthg
StepHypRef Expression
1 altopthsn 35466 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2 sneqbg 4839 . . 3 (𝐴𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3 sneqbg 4839 . . 3 (𝐵𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
42, 3bi2anan9 636 . 2 ((𝐴𝑉𝐵𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
51, 4bitrid 283 1 ((𝐴𝑉𝐵𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  {csn 4623  caltop 35461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-altop 35463
This theorem is referenced by:  altopth  35474
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