Theorem truxortru 1350

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 1307	. 2 ⊢ ((⊤ ⊻ ⊤) ↔ ((⊤ ∨ ⊤) ∧ ¬ (⊤ ∧ ⊤)))
2		oridm 706	. . 3 ⊢ ((⊤ ∨ ⊤) ↔ ⊤)
3		nottru 1344	. . . 4 ⊢ (¬ ⊤ ↔ ⊥)
4		anidm 388	. . . 4 ⊢ ((⊤ ∧ ⊤) ↔ ⊤)
5	3, 4	xchnxbir 638	. . 3 ⊢ (¬ (⊤ ∧ ⊤) ↔ ⊥)
6	2, 5	anbi12i 447	. 2 ⊢ (((⊤ ∨ ⊤) ∧ ¬ (⊤ ∧ ⊤)) ↔ (⊤ ∧ ⊥))
7		truan 1301	. 2 ⊢ ((⊤ ∧ ⊥) ↔ ⊥)
8	1, 6, 7	3bitri 204	1 ⊢ ((⊤ ⊻ ⊤) ↔ ⊥)

Syntax hints:  ¬ wn 3   ∧ wa 102   ↔ wb 103   ∨ wo 661  ⊤wtru 1285  ⊥wfal 1289   ⊻ wxo 1306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-xor 1307

This theorem is referenced by: (None)