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Theorem altopthb 35951
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 35950 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
Hypotheses
Ref Expression
altopthb.1 𝐴 ∈ V
altopthb.2 𝐷 ∈ V
Assertion
Ref Expression
altopthb (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem altopthb
StepHypRef Expression
1 altopthb.1 . 2 𝐴 ∈ V
2 altopthb.2 . 2 𝐷 ∈ V
3 altopthbg 35949 . 2 ((𝐴 ∈ V ∧ 𝐷 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
41, 2, 3mp2an 692 1 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  caltop 35937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-sn 4631  df-pr 4633  df-altop 35939
This theorem is referenced by:  altopthc  35952
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