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Theorem ifpim1 40175
Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpim1 ((𝜑𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))

Proof of Theorem ifpim1
StepHypRef Expression
1 tru 1542 . . . 4
21olci 863 . . 3 (¬ ¬ 𝜑 ∨ ⊤)
32biantrur 534 . 2 ((¬ 𝜑𝜓) ↔ ((¬ ¬ 𝜑 ∨ ⊤) ∧ (¬ 𝜑𝜓)))
4 imor 850 . 2 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
5 dfifp4 1062 . 2 (if-(¬ 𝜑, ⊤, 𝜓) ↔ ((¬ ¬ 𝜑 ∨ ⊤) ∧ (¬ 𝜑𝜓)))
63, 4, 53bitr4i 306 1 ((𝜑𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  if-wif 1058  wtru 1539
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 210  df-an 400  df-or 845  df-ifp 1059  df-tru 1541
This theorem is referenced by: (None)
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