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Theorem ifpim1 36832
 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 1479 . . . 4
21olci 405 . . 3 (¬ ¬ 𝜑 ∨ ⊤)
32biantrur 526 . 2 ((¬ 𝜑𝜓) ↔ ((¬ ¬ 𝜑 ∨ ⊤) ∧ (¬ 𝜑𝜓)))
4 imor 427 . 2 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
5 dfifp4 1010 . 2 (if-(¬ 𝜑, ⊤, 𝜓) ↔ ((¬ ¬ 𝜑 ∨ ⊤) ∧ (¬ 𝜑𝜓)))
63, 4, 53bitr4i 291 1 ((𝜑𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  if-wif 1006  ⊤wtru 1476 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 196  df-or 384  df-an 385  df-ifp 1007  df-tru 1478 This theorem is referenced by:  ifpdfbi  36837
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