Theorem xorcom 1503

Description: The connector ⊻ 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 224	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
2	1	notbii 322	. 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ¬ (𝜓 ↔ 𝜑))
3		df-xor 1501	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
4		df-xor 1501	. 2 ⊢ ((𝜓 ⊻ 𝜑) ↔ ¬ (𝜓 ↔ 𝜑))
5	2, 3, 4	3bitr4i 305	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 208   ⊻ wxo 1500

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 209  df-xor 1501

This theorem is referenced by:  xorneg1  1511  falxortru  1580  hadcomaOLD  1596  hadcomb  1597  cadcoma  1609