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Theorem ifpnim2 38338
Description: Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpnim2 (¬ (𝜑𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑))

Proof of Theorem ifpnim2
StepHypRef Expression
1 ifpnot23c 38323 . 2 (¬ if-(𝜓, 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜓, 𝜑))
2 ifpim4 38337 . 2 ((𝜑𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))
31, 2xchnxbir 322 1 (¬ (𝜑𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  if-wif 1050
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 197  df-or 384  df-an 385  df-ifp 1051
This theorem is referenced by: (None)
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