Theorem truxorfal 1415

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 1371	. 2 ⊢ ((⊤ ⊻ ⊥) ↔ ((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)))
2		truorfal 1401	. . 3 ⊢ ((⊤ ∨ ⊥) ↔ ⊤)
3		notfal 1409	. . . 4 ⊢ (¬ ⊥ ↔ ⊤)
4		truan 1365	. . . 4 ⊢ ((⊤ ∧ ⊥) ↔ ⊥)
5	3, 4	xchnxbir 676	. . 3 ⊢ (¬ (⊤ ∧ ⊥) ↔ ⊤)
6	2, 5	anbi12i 457	. 2 ⊢ (((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)) ↔ (⊤ ∧ ⊤))
7		anidm 394	. 2 ⊢ ((⊤ ∧ ⊤) ↔ ⊤)
8	1, 6, 7	3bitri 205	1 ⊢ ((⊤ ⊻ ⊥) ↔ ⊤)

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 703  ⊤wtru 1349  ⊥wfal 1353   ⊻ wxo 1370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-xor 1371

This theorem is referenced by: (None)