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Theorem nbn 347
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
Hypothesis
Ref Expression
nbn.1  |-  -.  ph
Assertion
Ref Expression
nbn  |-  ( -. 
ps 
<->  ( ps  <->  ph ) )

Proof of Theorem nbn
StepHypRef Expression
1 nbn.1 . . 3  |-  -.  ph
2 bibif 346 . . 3  |-  ( -. 
ph  ->  ( ( ps  <->  ph )  <->  -.  ps )
)
31, 2ax-mp 5 . 2  |-  ( ( ps  <->  ph )  <->  -.  ps )
43bicomi 202 1  |-  ( -. 
ps 
<->  ( ps  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184
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 185
This theorem is referenced by:  nbn3  348  nbfal  1375  n0f  3642  disj  3716  axnulALT  4416  dm0rn0  5052  reldm0  5053  isarchi  26132
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