Theorem nancom

Description: Alternative denial is commutative. Remark: alternative denial is not associative, see nanass . (Contributed by Mario Carneiro, 9-May-2015) (Proof shortened by Wolf Lammen, 26-Jun-2020)

	Ref	Expression
Assertion	nancom	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ( 𝜓 ⊼ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		con2b	⊢ ( ( 𝜑 → ¬ 𝜓 ) ↔ ( 𝜓 → ¬ 𝜑 ) )
2		dfnan2	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ( 𝜑 → ¬ 𝜓 ) )
3		dfnan2	⊢ ( ( 𝜓 ⊼ 𝜑 ) ↔ ( 𝜓 → ¬ 𝜑 ) )
4	1 2 3	3bitr4i	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ( 𝜓 ⊼ 𝜑 ) )