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Theorem ifpnim1 39853
Description: Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpnim1 (¬ (𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑))

Proof of Theorem ifpnim1
StepHypRef Expression
1 ifpnot23c 39840 . 2 (¬ if-(𝜑, 𝜓, ¬ 𝜑) ↔ if-(𝜑, ¬ 𝜓, 𝜑))
2 ifpim3 39852 . 2 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑))
31, 2xchnxbir 335 1 (¬ (𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  if-wif 1056
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 1057
This theorem is referenced by: (None)
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