Theorem ifpim4 44483

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 490	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2		olc 882	. 2 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
3		ifpim23g 44480	. 2 ⊢ (((𝜑 → 𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑)) ↔ (((𝜑 ∧ 𝜓) → 𝜓) ∧ (𝜓 → (𝜑 ∨ 𝜓))))
4	1, 2, 3	mpbir2an 724	1 ⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  ifpnim2  44484