Theorem ifpid2

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

	Ref	Expression
Assertion	ifpid2	⊢ ( 𝜑 ↔ if- ( 𝜑 , ⊤ , ⊥ ) )

Proof

Step	Hyp	Ref	Expression
1		tru	⊢ ⊤
2	1	olci	⊢ ( ¬ 𝜑 ∨ ⊤ )
3	2	biantrur	⊢ ( ( 𝜑 ∨ ⊥ ) ↔ ( ( ¬ 𝜑 ∨ ⊤ ) ∧ ( 𝜑 ∨ ⊥ ) ) )
4		fal	⊢ ¬ ⊥
5	4	biorfri	⊢ ( 𝜑 ↔ ( 𝜑 ∨ ⊥ ) )
6		dfifp4	⊢ ( if- ( 𝜑 , ⊤ , ⊥ ) ↔ ( ( ¬ 𝜑 ∨ ⊤ ) ∧ ( 𝜑 ∨ ⊥ ) ) )
7	3 5 6	3bitr4i	⊢ ( 𝜑 ↔ if- ( 𝜑 , ⊤ , ⊥ ) )