Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpdfan2 Structured version   Visualization version   GIF version

Theorem ifpdfan2 38329
Description: Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpdfan2 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜑))

Proof of Theorem ifpdfan2
StepHypRef Expression
1 id 22 . . . 4 (𝜑𝜑)
21notnoti 139 . . 3 ¬ ¬ (𝜑𝜑)
32biorfi 909 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜑)))
4 dfifp6 1055 . 2 (if-(𝜑, 𝜓, 𝜑) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜑)))
53, 4bitr4i 267 1 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  if-wif 1049
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-an 383  df-or 827  df-ifp 1050
This theorem is referenced by:  ifpancor  38330
  Copyright terms: Public domain W3C validator