Theorem truxorfal 1366

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 1322	. 2 ⊢ ((⊤ ⊻ ⊥) ↔ ((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)))
2		truorfal 1352	. . 3 ⊢ ((⊤ ∨ ⊥) ↔ ⊤)
3		notfal 1360	. . . 4 ⊢ (¬ ⊥ ↔ ⊤)
4		truan 1316	. . . 4 ⊢ ((⊤ ∧ ⊥) ↔ ⊥)
5	3, 4	xchnxbir 647	. . 3 ⊢ (¬ (⊤ ∧ ⊥) ↔ ⊤)
6	2, 5	anbi12i 451	. 2 ⊢ (((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)) ↔ (⊤ ∧ ⊤))
7		anidm 391	. 2 ⊢ ((⊤ ∧ ⊤) ↔ ⊤)
8	1, 6, 7	3bitri 205	1 ⊢ ((⊤ ⊻ ⊥) ↔ ⊤)

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 670  ⊤wtru 1300  ⊥wfal 1304   ⊻ wxo 1321

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 584  ax-in2 585  ax-io 671

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-xor 1322

This theorem is referenced by: (None)