MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  con2bi Structured version   Visualization version   GIF version

Theorem con2bi 342
Description: Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)
Assertion
Ref Expression
con2bi ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))

Proof of Theorem con2bi
StepHypRef Expression
1 notbi 308 . 2 ((𝜑 ↔ ¬ 𝜓) ↔ (¬ 𝜑 ↔ ¬ ¬ 𝜓))
2 notnotb 304 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
32bibi2i 326 . 2 ((¬ 𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ ¬ 𝜓))
4 bicom 212 . 2 ((¬ 𝜑𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
51, 3, 43bitr2i 288 1 ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196
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
This theorem is referenced by:  con2bid  343  nbbn  372
  Copyright terms: Public domain W3C validator