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Theorem altopthd 34370
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 34367 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 2743 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫)
2 altopthd.1 . . 3 𝐶 ∈ V
3 altopthd.2 . . 3 𝐷 ∈ V
42, 3altopth 34367 . 2 (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴𝐷 = 𝐵))
5 eqcom 2743 . . 3 (𝐶 = 𝐴𝐴 = 𝐶)
6 eqcom 2743 . . 3 (𝐷 = 𝐵𝐵 = 𝐷)
75, 6anbi12i 627 . 2 ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐴 = 𝐶𝐵 = 𝐷))
81, 4, 73bitri 296 1 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  caltop 34354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-sn 4574  df-pr 4576  df-altop 34356
This theorem is referenced by: (None)
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