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Theorem truxorfal 1311
 Description: A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
Assertion
Ref Expression
truxorfal ((⊤ ⊻ ⊥) ↔ ⊤)

Proof of Theorem truxorfal
StepHypRef Expression
1 df-xor 1267 . 2 ((⊤ ⊻ ⊥) ↔ ((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)))
2 truorfal 1297 . . 3 ((⊤ ∨ ⊥) ↔ ⊤)
3 notfal 1305 . . . 4 (¬ ⊥ ↔ ⊤)
4 truan 1260 . . . 4 ((⊤ ∧ ⊥) ↔ ⊥)
53, 4xchnxbir 606 . . 3 (¬ (⊤ ∧ ⊥) ↔ ⊤)
62, 5anbi12i 433 . 2 (((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)) ↔ (⊤ ∧ ⊤))
7 anidm 376 . 2 ((⊤ ∧ ⊤) ↔ ⊤)
81, 6, 73bitri 195 1 ((⊤ ⊻ ⊥) ↔ ⊤)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∨ wo 629  ⊤wtru 1244  ⊥wfal 1248   ⊻ wxo 1266 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 630 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-xor 1267 This theorem is referenced by: (None)
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