Theorem nanimn

Description: Alternative denial in terms of our primitive connectives (implication and negation). (Contributed by WL, 26-Jun-2020)

	Ref	Expression
Assertion	nanimn	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ( 𝜑 → ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		df-nan	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
2		imnan	⊢ ( ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
3	1 2	bitr4i	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ( 𝜑 → ¬ 𝜓 ) )