MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mt2bi Structured version   Visualization version   Unicode version

Theorem mt2bi 340
Description: A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
Hypothesis
Ref Expression
mt2bi.1  |-  ph
Assertion
Ref Expression
mt2bi  |-  ( -. 
ps 
<->  ( ps  ->  -.  ph ) )

Proof of Theorem mt2bi
StepHypRef Expression
1 mt2bi.1 . . 3  |-  ph
21a1bi 339 . 2  |-  ( -. 
ps 
<->  ( ph  ->  -.  ps ) )
3 con2b 336 . 2  |-  ( (
ph  ->  -.  ps )  <->  ( ps  ->  -.  ph )
)
42, 3bitri 253 1  |-  ( -. 
ps 
<->  ( ps  ->  -.  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188
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 189
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator