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Theorem ifpnot 38131
 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 1527 . . . 4
21olci 405 . . 3 (𝜑 ∨ ⊤)
32biantru 525 . 2 ((¬ 𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
4 fal 1530 . . 3 ¬ ⊥
54biorfi 421 . 2 𝜑 ↔ (¬ 𝜑 ∨ ⊥))
6 dfifp4 1036 . 2 (if-(𝜑, ⊥, ⊤) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
73, 5, 63bitr4i 292 1 𝜑 ↔ if-(𝜑, ⊥, ⊤))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 382   ∧ wa 383  if-wif 1032  ⊤wtru 1524  ⊥wfal 1528 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 384  df-an 385  df-ifp 1033  df-tru 1526  df-fal 1529 This theorem is referenced by: (None)
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