Theorem truxortru 1362

Description: A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)

Assertion

Ref	Expression
truxortru	⊢ ((⊤ ⊻ ⊤) ↔ ⊥)

Proof of Theorem truxortru

Step	Hyp	Ref	Expression
1		df-xor 1319	. 2 ⊢ ((⊤ ⊻ ⊤) ↔ ((⊤ ∨ ⊤) ∧ ¬ (⊤ ∧ ⊤)))
2		oridm 712	. . 3 ⊢ ((⊤ ∨ ⊤) ↔ ⊤)
3		nottru 1356	. . . 4 ⊢ (¬ ⊤ ↔ ⊥)
4		anidm 389	. . . 4 ⊢ ((⊤ ∧ ⊤) ↔ ⊤)
5	3, 4	xchnxbir 644	. . 3 ⊢ (¬ (⊤ ∧ ⊤) ↔ ⊥)
6	2, 5	anbi12i 449	. 2 ⊢ (((⊤ ∨ ⊤) ∧ ¬ (⊤ ∧ ⊤)) ↔ (⊤ ∧ ⊥))
7		truan 1313	. 2 ⊢ ((⊤ ∧ ⊥) ↔ ⊥)
8	1, 6, 7	3bitri 205	1 ⊢ ((⊤ ⊻ ⊤) ↔ ⊥)

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 667  ⊤wtru 1297  ⊥wfal 1301   ⊻ wxo 1318

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-xor 1319

This theorem is referenced by: (None)