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Theorem nbn 648
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 647 . . 3  |-  ( -. 
ph  ->  ( ( ps  <->  ph )  <->  -.  ps )
)
31, 2ax-mp 7 . 2  |-  ( ( ps  <->  ph )  <->  -.  ps )
43bicomi 130 1  |-  ( -. 
ps 
<->  ( ps  <->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> 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:  nbn3  649  nbfal  1296  n0rf  3267  eq0  3273  disj  3299  dm0rn0  4580  reldm0  4581
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