Theorem ifpnim1

Description: Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020)

	Ref	Expression
Assertion	ifpnim1	⊢ ( ¬ ( 𝜑 → 𝜓 ) ↔ if- ( 𝜑 , ¬ 𝜓 , 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ifpnot23c	⊢ ( ¬ if- ( 𝜑 , 𝜓 , ¬ 𝜑 ) ↔ if- ( 𝜑 , ¬ 𝜓 , 𝜑 ) )
2		ifpim3	⊢ ( ( 𝜑 → 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ¬ 𝜑 ) )
3	1 2	xchnxbir	⊢ ( ¬ ( 𝜑 → 𝜓 ) ↔ if- ( 𝜑 , ¬ 𝜓 , 𝜑 ) )