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Theorem ifpim2 37297
Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpim2 ((𝜑𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))

Proof of Theorem ifpim2
StepHypRef Expression
1 tru 1484 . . . 4
21olci 406 . . 3 𝜓 ∨ ⊤)
32biantrur 527 . 2 ((𝜓 ∨ ¬ 𝜑) ↔ ((¬ 𝜓 ∨ ⊤) ∧ (𝜓 ∨ ¬ 𝜑)))
4 imor 428 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
5 orcom 402 . . 3 ((¬ 𝜑𝜓) ↔ (𝜓 ∨ ¬ 𝜑))
64, 5bitri 264 . 2 ((𝜑𝜓) ↔ (𝜓 ∨ ¬ 𝜑))
7 dfifp4 1015 . 2 (if-(𝜓, ⊤, ¬ 𝜑) ↔ ((¬ 𝜓 ∨ ⊤) ∧ (𝜓 ∨ ¬ 𝜑)))
83, 6, 73bitr4i 292 1 ((𝜑𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  if-wif 1011  wtru 1481
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  df-tru 1483
This theorem is referenced by:  ifpdfbi  37299
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