Theorem ifpnot

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

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

Proof

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