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Theorem nbn2 646
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
nbn2  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )

Proof of Theorem nbn2
StepHypRef Expression
1 pm5.21im 645 . 2  |-  ( -. 
ph  ->  ( -.  ps  ->  ( ph  <->  ps )
) )
2 bi2 128 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
3 mtt 643 . . 3  |-  ( -. 
ph  ->  ( -.  ps  <->  ( ps  ->  ph ) ) )
42, 3syl5ibr 154 . 2  |-  ( -. 
ph  ->  ( ( ph  <->  ps )  ->  -.  ps )
)
51, 4impbid 127 1  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bibif  647  pm5.18dc  811  biassdc  1327
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