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Theorem altopthd 31249
 Description: Alternate ordered pair theorem with different sethood requirements. See altopth 31246 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
Hypotheses
Ref Expression
altopthd.1 𝐶 ∈ V
altopthd.2 𝐷 ∈ V
Assertion
Ref Expression
altopthd (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem altopthd
StepHypRef Expression
1 eqcom 2617 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫)
2 altopthd.1 . . 3 𝐶 ∈ V
3 altopthd.2 . . 3 𝐷 ∈ V
42, 3altopth 31246 . 2 (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴𝐷 = 𝐵))
5 eqcom 2617 . . 3 (𝐶 = 𝐴𝐴 = 𝐶)
6 eqcom 2617 . . 3 (𝐷 = 𝐵𝐵 = 𝐷)
75, 6anbi12i 729 . 2 ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐴 = 𝐶𝐵 = 𝐷))
81, 4, 73bitri 285 1 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟪caltop 31233 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-altop 31235 This theorem is referenced by: (None)
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