Theorem ifpim3 44036

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

Assertion

Ref	Expression
ifpim3	⊢ ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑))

Proof of Theorem ifpim3

Step	Hyp	Ref	Expression
1		simpl 486	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2		orc 878	. 2 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3		ifpim23g 44035	. 2 ⊢ (((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑)) ↔ (((𝜑 ∧ 𝜓) → 𝜑) ∧ (𝜑 → (𝜑 ∨ 𝜓))))
4	1, 2, 3	mpbir2an 721	1 ⊢ ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858  if-wif 1073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1074

This theorem is referenced by:  ifpnim1  44037