Theorem ifpim1

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

	Ref	Expression
Assertion	ifpim1	⊢ ( ( 𝜑 → 𝜓 ) ↔ if- ( ¬ 𝜑 , ⊤ , 𝜓 ) )

Proof

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