ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  truxorfal GIF version

Theorem truxorfal 1383
Description: A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
Assertion
Ref Expression
truxorfal ((⊤ ⊻ ⊥) ↔ ⊤)

Proof of Theorem truxorfal
StepHypRef Expression
1 df-xor 1339 . 2 ((⊤ ⊻ ⊥) ↔ ((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)))
2 truorfal 1369 . . 3 ((⊤ ∨ ⊥) ↔ ⊤)
3 notfal 1377 . . . 4 (¬ ⊥ ↔ ⊤)
4 truan 1333 . . . 4 ((⊤ ∧ ⊥) ↔ ⊥)
53, 4xchnxbir 655 . . 3 (¬ (⊤ ∧ ⊥) ↔ ⊤)
62, 5anbi12i 455 . 2 (((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)) ↔ (⊤ ∧ ⊤))
7 anidm 393 . 2 ((⊤ ∧ ⊤) ↔ ⊤)
81, 6, 73bitri 205 1 ((⊤ ⊻ ⊥) ↔ ⊤)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 682  wtru 1317  wfal 1321  wxo 1338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-xor 1339
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator