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Theorem altopthbg 32050
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 32043 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2 sneqbg 4365 . . 3 (𝐴𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3 sneqbg 4365 . . . 4 (𝐷𝑊 → ({𝐷} = {𝐵} ↔ 𝐷 = 𝐵))
4 eqcom 2627 . . . 4 ({𝐵} = {𝐷} ↔ {𝐷} = {𝐵})
5 eqcom 2627 . . . 4 (𝐵 = 𝐷𝐷 = 𝐵)
63, 4, 53bitr4g 303 . . 3 (𝐷𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
72, 6bi2anan9 916 . 2 ((𝐴𝑉𝐷𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
81, 7syl5bb 272 1 ((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  {csn 4168  caltop 32038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-sn 4169  df-pr 4171  df-altop 32040
This theorem is referenced by:  altopthb  32052
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