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Theorem altopth 31246
Description: The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that 𝐶 and 𝐷 are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4871), requires 𝐷 to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)
Hypotheses
Ref Expression
altopth.1 𝐴 ∈ V
altopth.2 𝐵 ∈ V
Assertion
Ref Expression
altopth (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem altopth
StepHypRef Expression
1 altopth.1 . 2 𝐴 ∈ V
2 altopth.2 . 2 𝐵 ∈ V
3 altopthg 31244 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
41, 2, 3mp2an 704 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:  altopthd  31249  altopelaltxp  31253
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