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

Theorem con2bi 320
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  |-  ( (
ph 
<->  -.  ps )  <->  ( ps  <->  -. 
ph ) )

Proof of Theorem con2bi
StepHypRef Expression
1 notbi 288 . 2  |-  ( (
ph 
<->  -.  ps )  <->  ( -.  ph  <->  -. 
-.  ps ) )
2 notnot 284 . . 3  |-  ( ps  <->  -. 
-.  ps )
32bibi2i 306 . 2  |-  ( ( -.  ph  <->  ps )  <->  ( -.  ph  <->  -. 
-.  ps ) )
4 bicom 193 . 2  |-  ( ( -.  ph  <->  ps )  <->  ( ps  <->  -. 
ph ) )
51, 3, 43bitr2i 266 1  |-  ( (
ph 
<->  -.  ps )  <->  ( ps  <->  -. 
ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178
This theorem is referenced by:  con2bid  321  nbbn  349
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179
  Copyright terms: Public domain W3C validator