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Theorem notbi 296
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 295 . 2  |-  ( (
ph 
<->  ps )  ->  ( -.  ph  <->  -.  ps )
)
3 id 22 . . 3  |-  ( ( -.  ph  <->  -.  ps )  ->  ( -.  ph  <->  -.  ps )
)
43con4bid 294 . 2  |-  ( ( -.  ph  <->  -.  ps )  ->  ( ph  <->  ps )
)
52, 4impbii 190 1  |-  ( (
ph 
<->  ps )  <->  ( -.  ph  <->  -. 
ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187
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 188
This theorem is referenced by:  notbii  297  con4bii  298  con2bi  329  nbn2  346  pm5.32  640  hadnot  1498  had0  1500  cbvexd  2091  symdifass  3645  isocnv3  6182  suppimacnv  6880  f1omvdco3  17033  onsuct0  31050  bj-cbvexdv  31242  ifpbi1  36034  ifpbi13  36046  abciffcbatnabciffncba  38331  abciffcbatnabciffncbai  38332
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