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Theorem naim1 24823
Description: Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
Assertion
Ref Expression
naim1  |-  ( (
ph  ->  ps )  -> 
( ( ps  -/\  ch )  ->  ( ph  -/\ 
ch ) ) )

Proof of Theorem naim1
StepHypRef Expression
1 con3 126 . . 3  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
21orim1d 812 . 2  |-  ( (
ph  ->  ps )  -> 
( ( -.  ps  \/  -.  ch )  -> 
( -.  ph  \/  -.  ch ) ) )
3 pm3.13 487 . . . 4  |-  ( -.  ( ps  /\  ch )  ->  ( -.  ps  \/  -.  ch ) )
4 pm3.14 488 . . . 4  |-  ( ( -.  ph  \/  -.  ch )  ->  -.  ( ph  /\  ch ) )
53, 4imim12i 53 . . 3  |-  ( ( ( -.  ps  \/  -.  ch )  ->  ( -.  ph  \/  -.  ch ) )  ->  ( -.  ( ps  /\  ch )  ->  -.  ( ph  /\ 
ch ) ) )
6 df-nan 1288 . . 3  |-  ( ( ps  -/\  ch )  <->  -.  ( ps  /\  ch ) )
7 df-nan 1288 . . 3  |-  ( (
ph  -/\  ch )  <->  -.  ( ph  /\  ch ) )
85, 6, 73imtr4g 261 . 2  |-  ( ( ( -.  ps  \/  -.  ch )  ->  ( -.  ph  \/  -.  ch ) )  ->  (
( ps  -/\  ch )  ->  ( ph  -/\  ch )
) )
92, 8syl 15 1  |-  ( (
ph  ->  ps )  -> 
( ( ps  -/\  ch )  ->  ( ph  -/\ 
ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    -/\ wnan 1287
This theorem is referenced by:  naim1i  24825
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-or 359  df-an 360  df-nan 1288
  Copyright terms: Public domain W3C validator