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Theorem ifpim2 40351
 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 1542 . . . 4
21olci 863 . . 3 𝜓 ∨ ⊤)
32biantrur 534 . 2 ((𝜓 ∨ ¬ 𝜑) ↔ ((¬ 𝜓 ∨ ⊤) ∧ (𝜓 ∨ ¬ 𝜑)))
4 imor 850 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
5 orcom 867 . . 3 ((¬ 𝜑𝜓) ↔ (𝜓 ∨ ¬ 𝜑))
64, 5bitri 278 . 2 ((𝜑𝜓) ↔ (𝜓 ∨ ¬ 𝜑))
7 dfifp4 1062 . 2 (if-(𝜓, ⊤, ¬ 𝜑) ↔ ((¬ 𝜓 ∨ ⊤) ∧ (𝜓 ∨ ¬ 𝜑)))
83, 6, 73bitr4i 306 1 ((𝜑𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))
 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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