Theorem ifpnim1 39856

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 39843	. 2 ⊢ (¬ if-(𝜑, 𝜓, ¬ 𝜑) ↔ if-(𝜑, ¬ 𝜓, 𝜑))
2		ifpim3 39855	. 2 ⊢ ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑))
3	1, 2	xchnxbir 335	1 ⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  if-wif 1057

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 399  df-or 844  df-ifp 1058

This theorem is referenced by: (None)