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Theorem altopthd 34882
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 34879 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 2740 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫)
2 altopthd.1 . . 3 𝐶 ∈ V
3 altopthd.2 . . 3 𝐷 ∈ V
42, 3altopth 34879 . 2 (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴𝐷 = 𝐵))
5 eqcom 2740 . . 3 (𝐶 = 𝐴𝐴 = 𝐶)
6 eqcom 2740 . . 3 (𝐷 = 𝐵𝐵 = 𝐷)
75, 6anbi12i 628 . 2 ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐴 = 𝐶𝐵 = 𝐷))
81, 4, 73bitri 297 1 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  caltop 34866
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 5298  ax-nul 5305  ax-pr 5426
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 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-altop 34868
This theorem is referenced by: (None)
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