Theorem truxorfal 1351

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

Assertion

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

Proof of Theorem truxorfal

Step	Hyp	Ref	Expression
1		df-xor 1307	. 2 ⊢ ((⊤ ⊻ ⊥) ↔ ((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)))
2		truorfal 1337	. . 3 ⊢ ((⊤ ∨ ⊥) ↔ ⊤)
3		notfal 1345	. . . 4 ⊢ (¬ ⊥ ↔ ⊤)
4		truan 1301	. . . 4 ⊢ ((⊤ ∧ ⊥) ↔ ⊥)
5	3, 4	xchnxbir 638	. . 3 ⊢ (¬ (⊤ ∧ ⊥) ↔ ⊤)
6	2, 5	anbi12i 447	. 2 ⊢ (((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)) ↔ (⊤ ∧ ⊤))
7		anidm 388	. 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)