Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpnot23b Structured version   Visualization version   GIF version

Theorem ifpnot23b 38327
Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpnot23b (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))

Proof of Theorem ifpnot23b
StepHypRef Expression
1 ifpnot23 38323 . 2 (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ 𝜒))
2 notnotb 306 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
3 ifpbi2 38311 . . 3 ((𝜓 ↔ ¬ ¬ 𝜓) → (if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ 𝜒)))
42, 3ax-mp 5 . 2 (if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ 𝜒))
51, 4bitr4i 269 1 (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  if-wif 1078
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
This theorem is referenced by:  ifpbiidcor2  38328
  Copyright terms: Public domain W3C validator