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Theorem ifpid2 38341
Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpid2 (𝜑 ↔ if-(𝜑, ⊤, ⊥))

Proof of Theorem ifpid2
StepHypRef Expression
1 tru 1635 . . . 4
21olci 853 . . 3 𝜑 ∨ ⊤)
32biantrur 520 . 2 ((𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ ⊥)))
4 fal 1638 . . 3 ¬ ⊥
54biorfi 922 . 2 (𝜑 ↔ (𝜑 ∨ ⊥))
6 dfifp4 1053 . 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  wa 382  wo 834  if-wif 1049  wtru 1632  wfal 1636
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-an 383  df-or 835  df-ifp 1050  df-tru 1634  df-fal 1637
This theorem is referenced by:  frege52aid  38678
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