Theorem ifpim4 41105

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

Assertion

Ref	Expression
ifpim4	⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))

Proof of Theorem ifpim4

Step	Hyp	Ref	Expression
1		simpr 485	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2		olc 865	. 2 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
3		ifpim23g 41102	. 2 ⊢ (((𝜑 → 𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑)) ↔ (((𝜑 ∧ 𝜓) → 𝜓) ∧ (𝜓 → (𝜑 ∨ 𝜓))))
4	1, 2, 3	mpbir2an 708	1 ⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  if-wif 1060

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 206  df-an 397  df-or 845  df-ifp 1061

This theorem is referenced by:  ifpnim2  41106