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Theorem symdifxor 3197
Description: Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
Assertion
Ref Expression
symdifxor ((AB) ∪ (BA)) = {x ∣ (x Ax B)}
Distinct variable groups:   x,A   x,B

Proof of Theorem symdifxor
StepHypRef Expression
1 eldif 2921 . . . 4 (x (AB) ↔ (x A ¬ x B))
2 eldif 2921 . . . 4 (x (BA) ↔ (x B ¬ x A))
31, 2orbi12i 680 . . 3 ((x (AB) x (BA)) ↔ ((x A ¬ x B) (x B ¬ x A)))
4 elun 3078 . . 3 (x ((AB) ∪ (BA)) ↔ (x (AB) x (BA)))
5 excxor 1268 . . . 4 ((x Ax B) ↔ ((x A ¬ x B) x A x B)))
6 ancom 253 . . . . 5 ((¬ x A x B) ↔ (x B ¬ x A))
76orbi2i 678 . . . 4 (((x A ¬ x B) x A x B)) ↔ ((x A ¬ x B) (x B ¬ x A)))
85, 7bitri 173 . . 3 ((x Ax B) ↔ ((x A ¬ x B) (x B ¬ x A)))
93, 4, 83bitr4i 201 . 2 (x ((AB) ∪ (BA)) ↔ (x Ax B))
109abbi2i 2149 1 ((AB) ∪ (BA)) = {x ∣ (x Ax B)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 628   = wceq 1242  wxo 1265   wcel 1390  {cab 2023  cdif 2908  cun 2909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-xor 1266  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916
This theorem is referenced by: (None)
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