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Theorem symdif2 3520
 Description: Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
symdif2 ((A B) ∪ (B A)) = {x ¬ (x Ax B)}
Distinct variable groups:   x,A   x,B

Proof of Theorem symdif2
StepHypRef Expression
1 eldif 3221 . . . 4 (x (A B) ↔ (x A ¬ x B))
2 eldif 3221 . . . 4 (x (B A) ↔ (x B ¬ x A))
31, 2orbi12i 507 . . 3 ((x (A B) x (B A)) ↔ ((x A ¬ x B) (x B ¬ x A)))
4 elun 3220 . . 3 (x ((A B) ∪ (B A)) ↔ (x (A B) x (B A)))
5 xor 861 . . 3 (¬ (x Ax B) ↔ ((x A ¬ x B) (x B ¬ x A)))
63, 4, 53bitr4i 268 . 2 (x ((A B) ∪ (B A)) ↔ ¬ (x Ax B))
76abbi2i 2464 1 ((A B) ∪ (B A)) = {x ¬ (x Ax B)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   ∖ cdif 3206   ∪ cun 3207 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215 This theorem is referenced by: (None)
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