Theorem xorcom

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

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

Proof

Step	Hyp	Ref	Expression
1		df-xor	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) )
2		bicom	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜓 ↔ 𝜑 ) )
3	1 2	xchbinx	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜓 ↔ 𝜑 ) )
4		df-xor	⊢ ( ( 𝜓 ⊻ 𝜑 ) ↔ ¬ ( 𝜓 ↔ 𝜑 ) )
5	3 4	bitr4i	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( 𝜓 ⊻ 𝜑 ) )