Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpnot Structured version   Visualization version   GIF version

Theorem ifpnot 36833
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 1479 . . . 4
21olci 405 . . 3 (𝜑 ∨ ⊤)
32biantru 525 . 2 ((¬ 𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
4 fal 1482 . . 3 ¬ ⊥
54biorfi 421 . 2 𝜑 ↔ (¬ 𝜑 ∨ ⊥))
6 dfifp4 1010 . 2 (if-(𝜑, ⊥, ⊤) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
73, 5, 63bitr4i 291 1 𝜑 ↔ if-(𝜑, ⊥, ⊤))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wo 382  wa 383  if-wif 1006  wtru 1476  wfal 1480
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 196  df-or 384  df-an 385  df-ifp 1007  df-tru 1478  df-fal 1481
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator