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Theorem ifpim4 37321
Description: Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpim4 ((𝜑𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))

Proof of Theorem ifpim4
StepHypRef Expression
1 simpr 477 . 2 ((𝜑𝜓) → 𝜓)
2 olc 399 . 2 (𝜓 → (𝜑𝜓))
3 ifpim23g 37318 . 2 (((𝜑𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑)) ↔ (((𝜑𝜓) → 𝜓) ∧ (𝜓 → (𝜑𝜓))))
41, 2, 3mpbir2an 954 1 ((𝜑𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  if-wif 1011
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 385  df-an 386  df-ifp 1012
This theorem is referenced by:  ifpnim2  37322
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