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Theorem ifpdfan 38304
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 1652 . . . 4 ¬ ⊥
21intnan 476 . . 3 ¬ (¬ 𝜑 ∧ ⊥)
32biorfi 953 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ⊥)))
4 df-ifp 1079 . 2 (if-(𝜑, 𝜓, ⊥) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ⊥)))
53, 4bitr4i 269 1 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  wo 865  if-wif 1078  wfal 1650
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 198  df-an 385  df-or 866  df-ifp 1079  df-tru 1641  df-fal 1651
This theorem is referenced by:  ifpdfnan  38325  ifpdfxor  38326
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