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Theorem ifpnot 39231
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 1511 . . . 4
21olci 852 . . 3 (𝜑 ∨ ⊤)
32biantru 522 . 2 ((¬ 𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
4 fal 1521 . . 3 ¬ ⊥
54biorfi 922 . 2 𝜑 ↔ (¬ 𝜑 ∨ ⊥))
6 dfifp4 1047 . 2 (if-(𝜑, ⊥, ⊤) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
73, 5, 63bitr4i 295 1 𝜑 ↔ if-(𝜑, ⊥, ⊤))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 387  wo 833  if-wif 1043  wtru 1508  wfal 1519
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 199  df-an 388  df-or 834  df-ifp 1044  df-tru 1510  df-fal 1520
This theorem is referenced by: (None)
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