Theorem ifpim3

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

	Ref	Expression
Assertion	ifpim3	⊢ ( ( 𝜑 → 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
2		orc	⊢ ( 𝜑 → ( 𝜑 ∨ 𝜓 ) )
3		ifpim23g	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ¬ 𝜑 ) ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) ∧ ( 𝜑 → ( 𝜑 ∨ 𝜓 ) ) ) )
4	1 2 3	mpbir2an	⊢ ( ( 𝜑 → 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ¬ 𝜑 ) )