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

Theorem nabctnabc 38238
Description: not ( a -> ( b /\ c ) ) we can show: not a implies ( b /\ c ) (Contributed by Jarvin Udandy, 7-Sep-2020.)
Hypothesis
Ref Expression
nabctnabc.1  |-  -.  ( ph  ->  ( ps  /\  ch ) )
Assertion
Ref Expression
nabctnabc  |-  ( -. 
ph  ->  ( ps  /\  ch ) )

Proof of Theorem nabctnabc
StepHypRef Expression
1 nabctnabc.1 . . . . . . . 8  |-  -.  ( ph  ->  ( ps  /\  ch ) )
2 pm4.61 428 . . . . . . . . 9  |-  ( -.  ( ph  ->  ( ps  /\  ch ) )  <-> 
( ph  /\  -.  ( ps  /\  ch ) ) )
32biimpi 198 . . . . . . . 8  |-  ( -.  ( ph  ->  ( ps  /\  ch ) )  ->  ( ph  /\  -.  ( ps  /\  ch ) ) )
41, 3ax-mp 5 . . . . . . 7  |-  ( ph  /\ 
-.  ( ps  /\  ch ) )
54simpli 460 . . . . . 6  |-  ph
64simpri 464 . . . . . 6  |-  -.  ( ps  /\  ch )
75, 62th 243 . . . . 5  |-  ( ph  <->  -.  ( ps  /\  ch ) )
8 bicom 204 . . . . . 6  |-  ( (
ph 
<->  -.  ( ps  /\  ch ) )  <->  ( -.  ( ps  /\  ch )  <->  ph ) )
98biimpi 198 . . . . 5  |-  ( (
ph 
<->  -.  ( ps  /\  ch ) )  ->  ( -.  ( ps  /\  ch ) 
<-> 
ph ) )
107, 9ax-mp 5 . . . 4  |-  ( -.  ( ps  /\  ch ) 
<-> 
ph )
1110biimpi 198 . . 3  |-  ( -.  ( ps  /\  ch )  ->  ph )
1211con3i 141 . 2  |-  ( -. 
ph  ->  -.  -.  ( ps  /\  ch ) )
1312notnotrd 117 1  |-  ( -. 
ph  ->  ( ps  /\  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371
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  df-an 373
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator