Users' Mathboxes Mathbox for Jarvin Udandy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aibnbna Unicode version

Theorem aibnbna 27874
Description: Given a implies b, not b, there exists a proof for not a. (Contributed by Jarvin Udandy, 1-Sep-2016.)
Hypotheses
Ref Expression
aibnbna.1  |-  ( ph  ->  ps )
aibnbna.2  |-  -.  ps
Assertion
Ref Expression
aibnbna  |-  -.  ph

Proof of Theorem aibnbna
StepHypRef Expression
1 aibnbna.1 . . . . 5  |-  ( ph  ->  ps )
2 ax-1 5 . . . . . . 7  |-  ( -. 
ps  ->  ( -.  ph  ->  -.  ps ) )
3 aibnbna.2 . . . . . . . 8  |-  -.  ps
43, 2mp1i 11 . . . . . . 7  |-  ( ( -.  ps  ->  ( -.  ph  ->  -.  ps )
)  ->  ( -.  ph 
->  -.  ps ) )
52, 4ax-mp 8 . . . . . 6  |-  ( -. 
ph  ->  -.  ps )
6 ax-3 7 . . . . . 6  |-  ( ( -.  ph  ->  -.  ps )  ->  ( ps  ->  ph ) )
75, 6ax-mp 8 . . . . 5  |-  ( ps 
->  ph )
81, 7pm3.2i 441 . . . 4  |-  ( (
ph  ->  ps )  /\  ( ps  ->  ph )
)
9 dfbi2 609 . . . . . 6  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
10 bicom 191 . . . . . . 7  |-  ( ( ( ph  <->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )  <->  ( (
( ph  ->  ps )  /\  ( ps  ->  ph )
)  <->  ( ph  <->  ps )
) )
1110biimpi 186 . . . . . 6  |-  ( ( ( ph  <->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )  -> 
( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( ph  <->  ps ) ) )
129, 11ax-mp 8 . . . . 5  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( ph  <->  ps ) )
1312biimpi 186 . . . 4  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  ( ph 
<->  ps ) )
148, 13ax-mp 8 . . 3  |-  ( ph  <->  ps )
15 nbfal 1316 . . . . 5  |-  ( -. 
ps 
<->  ( ps  <->  F.  )
)
1615biimpi 186 . . . 4  |-  ( -. 
ps  ->  ( ps  <->  F.  )
)
173, 16mp1i 11 . . . 4  |-  ( ( -.  ps  ->  ( ps 
<->  F.  ) )  -> 
( ps  <->  F.  )
)
1816, 17ax-mp 8 . . 3  |-  ( ps  <->  F.  )
1914, 18aisbbisfaisf 27870 . 2  |-  ( ph  <->  F.  )
2019aisfina 27866 1  |-  -.  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    F. wfal 1308
This theorem is referenced by:  aibnbaif  27875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-fal 1311
  Copyright terms: Public domain W3C validator