Theorem falxorfal 1383

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

Assertion

Ref	Expression
falxorfal	⊢ ((⊥ ⊻ ⊥) ↔ ⊥)

Proof of Theorem falxorfal

Step	Hyp	Ref	Expression
1		df-xor 1337	. 2 ⊢ ((⊥ ⊻ ⊥) ↔ ((⊥ ∨ ⊥) ∧ ¬ (⊥ ∧ ⊥)))
2		oridm 729	. . 3 ⊢ ((⊥ ∨ ⊥) ↔ ⊥)
3		notfal 1375	. . . 4 ⊢ (¬ ⊥ ↔ ⊤)
4		anidm 391	. . . 4 ⊢ ((⊥ ∧ ⊥) ↔ ⊥)
5	3, 4	xchnxbir 653	. . 3 ⊢ (¬ (⊥ ∧ ⊥) ↔ ⊤)
6	2, 5	anbi12i 453	. 2 ⊢ (((⊥ ∨ ⊥) ∧ ¬ (⊥ ∧ ⊥)) ↔ (⊥ ∧ ⊤))
7		falantru 1364	. 2 ⊢ ((⊥ ∧ ⊤) ↔ ⊥)
8	1, 6, 7	3bitri 205	1 ⊢ ((⊥ ⊻ ⊥) ↔ ⊥)

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 680  ⊤wtru 1315  ⊥wfal 1319   ⊻ wxo 1336

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 586  ax-in2 587  ax-io 681

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-xor 1337

This theorem is referenced by: (None)