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Theorem symdifxor 3246
Description: Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
Assertion
Ref Expression
symdifxor ((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem symdifxor
StepHypRef Expression
1 eldif 2991 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 2991 . . . 4 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
31, 2orbi12i 714 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
4 elun 3123 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)))
5 excxor 1310 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (¬ 𝑥𝐴𝑥𝐵)))
6 ancom 262 . . . . 5 ((¬ 𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
76orbi2i 712 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (¬ 𝑥𝐴𝑥𝐵)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
85, 7bitri 182 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
93, 4, 83bitr4i 210 . 2 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (𝑥𝐴𝑥𝐵))
109abbi2i 2197 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wo 662   = wceq 1285  wxo 1307  wcel 1434  {cab 2069  cdif 2979  cun 2980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-xor 1308  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-un 2986
This theorem is referenced by: (None)
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