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Theorem ifpid2 40166
 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 1542 . . . 4
21olci 863 . . 3 𝜑 ∨ ⊤)
32biantrur 534 . 2 ((𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ ⊥)))
4 fal 1552 . . 3 ¬ ⊥
54biorfi 936 . 2 (𝜑 ↔ (𝜑 ∨ ⊥))
6 dfifp4 1062 . 2 (if-(𝜑, ⊤, ⊥) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ ⊥)))
73, 5, 63bitr4i 306 1 (𝜑 ↔ if-(𝜑, ⊤, ⊥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058  ⊤wtru 1539  ⊥wfal 1550 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 210  df-an 400  df-or 845  df-ifp 1059  df-tru 1541  df-fal 1551 This theorem is referenced by:  frege52aid  40546
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