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Theorem ifpdfan 42775
Description: Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpdfan ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊥))

Proof of Theorem ifpdfan
StepHypRef Expression
1 fal 1547 . . . 4 ¬ ⊥
21intnan 486 . . 3 ¬ (¬ 𝜑 ∧ ⊥)
32biorfi 935 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ⊥)))
4 df-ifp 1060 . 2 (if-(𝜑, 𝜓, ⊥) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ⊥)))
53, 4bitr4i 278 1 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wo 844  if-wif 1059  wfal 1545
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 206  df-an 396  df-or 845  df-ifp 1060  df-tru 1536  df-fal 1546
This theorem is referenced by:  ifpdfnan  42795  ifpdfxor  42796
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