Theorem ifpnim1 44251

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

Assertion

Ref	Expression
ifpnim1	⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑))

Proof of Theorem ifpnim1

Step	Hyp	Ref	Expression
1		ifpnot23c 44238	. 2 ⊢ (¬ if-(𝜑, 𝜓, ¬ 𝜑) ↔ if-(𝜑, ¬ 𝜓, 𝜑))
2		ifpim3 44250	. 2 ⊢ ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑))
3	1, 2	xchnxbir 336	1 ⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  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 401  df-or 861  df-ifp 1079

This theorem is used by: (None)