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Theorem symdif2OLD 4054
Description: Obsolete version of dfsymdif4 4047 as of 18-Aug-2022. Two ways to express symmetric difference (df-symdif 4041). (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
symdif2OLD ((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem symdif2OLD
StepHypRef Expression
1 eldif 3779 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 3779 . . . 4 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
31, 2orbi12i 939 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
4 elun 3951 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)))
5 xor 1039 . . 3 (¬ (𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
63, 4, 53bitr4i 295 . 2 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ ¬ (𝑥𝐴𝑥𝐵))
76abbi2i 2915 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 385  wo 874   = wceq 1653  wcel 2157  {cab 2785  cdif 3766  cun 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-un 3774
This theorem is referenced by: (None)
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