MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bibif Structured version   Visualization version   GIF version

Theorem bibif 371
Description: Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
Assertion
Ref Expression
bibif 𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Proof of Theorem bibif
StepHypRef Expression
1 nbn2 370 . 2 𝜓 → (¬ 𝜑 ↔ (𝜓𝜑)))
2 bicom 221 . 2 ((𝜓𝜑) ↔ (𝜑𝜓))
31, 2bitr2di 287 1 𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205
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
This theorem is referenced by:  nbn  372  bj-bibibi  34796  or3or  41655
  Copyright terms: Public domain W3C validator