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Theorem symdifxor 3413
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 3150 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 3150 . . . 4 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
31, 2orbi12i 765 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
4 elun 3288 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)))
5 excxor 1388 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (¬ 𝑥𝐴𝑥𝐵)))
6 ancom 266 . . . . 5 ((¬ 𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
76orbi2i 763 . . . 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 2302 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wo 709   = wceq 1363  wxo 1385  wcel 2158  {cab 2173  cdif 3138  cun 3139
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-xor 1386  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145
This theorem is referenced by: (None)
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