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Theorem notbi 293
Description: Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.)
Assertion
Ref Expression
notbi  |-  ( (
ph 
<->  ps )  <->  ( -.  ph  <->  -. 
ps ) )

Proof of Theorem notbi
StepHypRef Expression
1 id 22 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ph 
<->  ps ) )
21notbid 292 . 2  |-  ( (
ph 
<->  ps )  ->  ( -.  ph  <->  -.  ps )
)
3 id 22 . . 3  |-  ( ( -.  ph  <->  -.  ps )  ->  ( -.  ph  <->  -.  ps )
)
43con4bid 291 . 2  |-  ( ( -.  ph  <->  -.  ps )  ->  ( ph  <->  ps )
)
52, 4impbii 188 1  |-  ( (
ph 
<->  ps )  <->  ( -.  ph  <->  -. 
ps ) )
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:  notbii  294  con4bii  295  con2bi  326  nbn2  343  pm5.32  634  cbvexd  2031  symdifass  3724  isocnv3  6203  suppimacnv  6902  f1omvdco3  16673  onsuct0  30134  bj-cbvexdv  34702  ifpbi1  38096  ifpbi13  38100
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