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Theorem atbiffatnnb 27984
Description: If a implies b, then a implies not not b (Contributed by Jarvin Udandy, 28-Aug-2016.)
Assertion
Ref Expression
atbiffatnnb  |-  ( (
ph  ->  ps )  -> 
( ph  ->  -.  -.  ps ) )

Proof of Theorem atbiffatnnb
StepHypRef Expression
1 idd 21 . . 3  |-  ( ph  ->  ( ps  ->  ps ) )
2 notnot 282 . . . . 5  |-  ( ps  <->  -. 
-.  ps )
3 bi1 178 . . . . 5  |-  ( ( ps  <->  -.  -.  ps )  ->  ( ps  ->  -.  -.  ps ) )
42, 3ax-mp 8 . . . 4  |-  ( ps 
->  -.  -.  ps )
54a1i 10 . . 3  |-  ( ( ps  ->  ps )  ->  ( ps  ->  -.  -.  ps ) )
61, 5syl 15 . 2  |-  ( ph  ->  ( ps  ->  -.  -.  ps ) )
76a2i 12 1  |-  ( (
ph  ->  ps )  -> 
( ph  ->  -.  -.  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176
This theorem is referenced by:  conimpf  27989
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
  Copyright terms: Public domain W3C validator