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Theorem bj-nnsn 12930
Description: As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-nnsn  |-  ( (
ph  ->  -.  ps )  <->  ( -.  -.  ph  ->  -. 
ps ) )

Proof of Theorem bj-nnsn
StepHypRef Expression
1 con3 631 . . . 4  |-  ( (
ph  ->  -.  ps )  ->  ( -.  -.  ps  ->  -.  ph ) )
21con3d 620 . . 3  |-  ( (
ph  ->  -.  ps )  ->  ( -.  -.  ph  ->  -.  -.  -.  ps ) )
3 notnotnot 623 . . 3  |-  ( -. 
-.  -.  ps  <->  -.  ps )
42, 3syl6ib 160 . 2  |-  ( (
ph  ->  -.  ps )  ->  ( -.  -.  ph  ->  -.  ps ) )
5 notnot 618 . . 3  |-  ( ph  ->  -.  -.  ph )
65imim1i 60 . 2  |-  ( ( -.  -.  ph  ->  -. 
ps )  ->  ( ph  ->  -.  ps )
)
74, 6impbii 125 1  |-  ( (
ph  ->  -.  ps )  <->  ( -.  -.  ph  ->  -. 
ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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