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Theorem ifpid2 37293
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 1484 . . . 4
21olci 406 . . 3 𝜑 ∨ ⊤)
32biantrur 527 . 2 ((𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ ⊥)))
4 fal 1487 . . 3 ¬ ⊥
54biorfi 422 . 2 (𝜑 ↔ (𝜑 ∨ ⊥))
6 dfifp4 1015 . 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 383  wa 384  if-wif 1011  wtru 1481  wfal 1485
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 385  df-an 386  df-ifp 1012  df-tru 1483  df-fal 1486
This theorem is referenced by:  frege52aid  37631
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