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Theorem altopthc 34938
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 34936 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
Hypotheses
Ref Expression
altopthc.1 𝐵 ∈ V
altopthc.2 𝐶 ∈ V
Assertion
Ref Expression
altopthc (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem altopthc
StepHypRef Expression
1 eqcom 2739 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫)
2 altopthc.2 . . 3 𝐶 ∈ V
3 altopthc.1 . . 3 𝐵 ∈ V
42, 3altopthb 34937 . 2 (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴𝐷 = 𝐵))
5 eqcom 2739 . . 3 (𝐶 = 𝐴𝐴 = 𝐶)
6 eqcom 2739 . . 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 1541  wcel 2106  Vcvv 3474  caltop 34923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-altop 34925
This theorem is referenced by: (None)
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