Theorem truxortru 1461

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 1418	. 2 ⊢ ((⊤ ⊻ ⊤) ↔ ((⊤ ∨ ⊤) ∧ ¬ (⊤ ∧ ⊤)))
2		oridm 762	. . 3 ⊢ ((⊤ ∨ ⊤) ↔ ⊤)
3		nottru 1455	. . . 4 ⊢ (¬ ⊤ ↔ ⊥)
4		anidm 396	. . . 4 ⊢ ((⊤ ∧ ⊤) ↔ ⊤)
5	3, 4	xchnxbir 685	. . 3 ⊢ (¬ (⊤ ∧ ⊤) ↔ ⊥)
6	2, 5	anbi12i 460	. 2 ⊢ (((⊤ ∨ ⊤) ∧ ¬ (⊤ ∧ ⊤)) ↔ (⊤ ∧ ⊥))
7		truan 1412	. 2 ⊢ ((⊤ ∧ ⊥) ↔ ⊥)
8	1, 6, 7	3bitri 206	1 ⊢ ((⊤ ⊻ ⊤) ↔ ⊥)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 713  ⊤wtru 1396  ⊥wfal 1400   ⊻ wxo 1417

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-xor 1418

This theorem is referenced by: (None)