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Theorem ifpdfan 40217
 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 1552 . . . 4 ¬ ⊥
21intnan 490 . . 3 ¬ (¬ 𝜑 ∧ ⊥)
32biorfi 936 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ⊥)))
4 df-ifp 1059 . 2 (if-(𝜑, 𝜓, ⊥) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ⊥)))
53, 4bitr4i 281 1 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058  ⊥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:  ifpdfnan  40237  ifpdfxor  40238
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