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Theorem altopeq12 35235
Description: Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopeq12 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)

Proof of Theorem altopeq12
StepHypRef Expression
1 sneq 4639 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
2 sneq 4639 . . 3 (𝐶 = 𝐷 → {𝐶} = {𝐷})
31, 2anim12i 612 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
4 altopthsn 35234 . 2 (⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫ ↔ ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
53, 4sylibr 233 1 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  {csn 4629  caltop 35229
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 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-altop 35231
This theorem is referenced by:  altopeq1  35236  altopeq2  35237
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