Theorem ifpim2

Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020)

	Ref	Expression
Assertion	ifpim2	⊢ ( ( 𝜑 → 𝜓 ) ↔ if- ( 𝜓 , ⊤ , ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		tru	⊢ ⊤
2	1	olci	⊢ ( ¬ 𝜓 ∨ ⊤ )
3	2	biantrur	⊢ ( ( 𝜓 ∨ ¬ 𝜑 ) ↔ ( ( ¬ 𝜓 ∨ ⊤ ) ∧ ( 𝜓 ∨ ¬ 𝜑 ) ) )
4		imor	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 ∨ 𝜓 ) )
5		orcom	⊢ ( ( ¬ 𝜑 ∨ 𝜓 ) ↔ ( 𝜓 ∨ ¬ 𝜑 ) )
6	4 5	bitri	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜓 ∨ ¬ 𝜑 ) )
7		dfifp4	⊢ ( if- ( 𝜓 , ⊤ , ¬ 𝜑 ) ↔ ( ( ¬ 𝜓 ∨ ⊤ ) ∧ ( 𝜓 ∨ ¬ 𝜑 ) ) )
8	3 6 7	3bitr4i	⊢ ( ( 𝜑 → 𝜓 ) ↔ if- ( 𝜓 , ⊤ , ¬ 𝜑 ) )