Theorem ifpim4

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

	Ref	Expression
Assertion	ifpim4	⊢ ( ( 𝜑 → 𝜓 ) ↔ if- ( 𝜓 , 𝜓 , ¬ 𝜑 ) )

Proof

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