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Theorem ifpnot 40171
Description: Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpnot 𝜑 ↔ if-(𝜑, ⊥, ⊤))

Proof of Theorem ifpnot
StepHypRef Expression
1 tru 1542 . . . 4
21olci 863 . . 3 (𝜑 ∨ ⊤)
32biantru 533 . 2 ((¬ 𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
4 fal 1552 . . 3 ¬ ⊥
54biorfi 936 . 2 𝜑 ↔ (¬ 𝜑 ∨ ⊥))
6 dfifp4 1062 . 2 (if-(𝜑, ⊥, ⊤) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
73, 5, 63bitr4i 306 1 𝜑 ↔ if-(𝜑, ⊥, ⊤))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wo 844  if-wif 1058  wtru 1539  wfal 1550
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  df-fal 1551
This theorem is referenced by: (None)
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