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Theorem altopeq12 35943
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 4640 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
2 sneq 4640 . . 3 (𝐶 = 𝐷 → {𝐶} = {𝐷})
31, 2anim12i 613 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
4 altopthsn 35942 . 2 (⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫ ↔ ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
53, 4sylibr 234 1 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  {csn 4630  caltop 35937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-sn 4631  df-pr 4633  df-altop 35939
This theorem is referenced by:  altopeq1  35944  altopeq2  35945
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