Theorem xorcom 1307

Description: ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
xorcom	⊢ ((φ ⊻ ψ) ↔ (ψ ⊻ φ))

Proof of Theorem xorcom

Step	Hyp	Ref	Expression
1		bicom 191	. . 3 ⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ))
2	1	notbii 287	. 2 ⊢ (¬ (φ ↔ ψ) ↔ ¬ (ψ ↔ φ))
3		df-xor 1305	. 2 ⊢ ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ))
4		df-xor 1305	. 2 ⊢ ((ψ ⊻ φ) ↔ ¬ (ψ ↔ φ))
5	2, 3, 4	3bitr4i 268	1 ⊢ ((φ ⊻ ψ) ↔ (ψ ⊻ φ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ⊻ wxo 1304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-xor 1305

This theorem is referenced by:  xorneg2  1312  hadcoma  1388  hadcomb  1389  cadcoma  1395