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Theorem symdifxor 3403
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 3140 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 3140 . . . 4 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
31, 2orbi12i 764 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
4 elun 3278 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)))
5 excxor 1378 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (¬ 𝑥𝐴𝑥𝐵)))
6 ancom 266 . . . . 5 ((¬ 𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
76orbi2i 762 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (¬ 𝑥𝐴𝑥𝐵)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
85, 7bitri 184 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
93, 4, 83bitr4i 212 . 2 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (𝑥𝐴𝑥𝐵))
109abbi2i 2292 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wo 708   = wceq 1353  wxo 1375  wcel 2148  {cab 2163  cdif 3128  cun 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-xor 1376  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135
This theorem is referenced by: (None)
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