Theorem truxorfal 1311

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 1267	. 2 ⊢ ((⊤ ⊻ ⊥) ↔ ((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)))
2		truorfal 1297	. . 3 ⊢ ((⊤ ∨ ⊥) ↔ ⊤)
3		notfal 1305	. . . 4 ⊢ (¬ ⊥ ↔ ⊤)
4		truan 1260	. . . 4 ⊢ ((⊤ ∧ ⊥) ↔ ⊥)
5	3, 4	xchnxbir 606	. . 3 ⊢ (¬ (⊤ ∧ ⊥) ↔ ⊤)
6	2, 5	anbi12i 433	. 2 ⊢ (((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)) ↔ (⊤ ∧ ⊤))
7		anidm 376	. 2 ⊢ ((⊤ ∧ ⊤) ↔ ⊤)
8	1, 6, 7	3bitri 195	1 ⊢ ((⊤ ⊻ ⊥) ↔ ⊤)

Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∨ wo 629  ⊤wtru 1244  ⊥wfal 1248   ⊻ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-xor 1267

This theorem is referenced by: (None)