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Theorem altopth1 35432
Description: Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopth1 (𝐴𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))

Proof of Theorem altopth1
StepHypRef Expression
1 altopthsn 35428 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2 sneqrg 4832 . . 3 (𝐴𝑉 → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
32adantrd 491 . 2 (𝐴𝑉 → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → 𝐴 = 𝐶))
41, 3biimtrid 241 1 (𝐴𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {csn 4620  caltop 35423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-sn 4621  df-pr 4623  df-altop 35425
This theorem is referenced by: (None)
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