Theorem xorneg1

Description: The connector \/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 27-Jun-2020)

	Ref	Expression
Assertion	xorneg1	⊢ ( ( ¬ 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜑 ⊻ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		xorcom	⊢ ( ( ¬ 𝜑 ⊻ 𝜓 ) ↔ ( 𝜓 ⊻ ¬ 𝜑 ) )
2		xorneg2	⊢ ( ( 𝜓 ⊻ ¬ 𝜑 ) ↔ ¬ ( 𝜓 ⊻ 𝜑 ) )
3		xorcom	⊢ ( ( 𝜓 ⊻ 𝜑 ) ↔ ( 𝜑 ⊻ 𝜓 ) )
4	2 3	xchbinx	⊢ ( ( 𝜓 ⊻ ¬ 𝜑 ) ↔ ¬ ( 𝜑 ⊻ 𝜓 ) )
5	1 4	bitri	⊢ ( ( ¬ 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜑 ⊻ 𝜓 ) )