Theorem xorcom 1509

Description: The connector ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)

Assertion

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

Proof of Theorem xorcom

Step	Hyp	Ref	Expression
1		df-xor 1507	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
2		bicom 221	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
3	1, 2	xchbinx 334	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜓 ↔ 𝜑))
4		df-xor 1507	. 2 ⊢ ((𝜓 ⊻ 𝜑) ↔ ¬ (𝜓 ↔ 𝜑))
5	3, 4	bitr4i 277	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 205   ⊻ wxo 1506

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 206  df-xor 1507

This theorem is referenced by:  xorneg1  1518  falxortru  1586  hadcomaOLD  1601  hadcomb  1602  cadcoma  1614