Metamath Proof Explorer


Theorem dfnan2

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

Ref Expression
Assertion dfnan2 ( ( 𝜑𝜓 ) ↔ ( 𝜑 → ¬ 𝜓 ) )

Proof

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