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Theorem con3th 924
Description: Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 126 demonstrates the use of the weak deduction theorem dedt 923 to derive it from con3i 127. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.)
Assertion
Ref Expression
con3th  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )

Proof of Theorem con3th
StepHypRef Expression
1 id 19 . . . 4  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ps  <->  ( ( ps  /\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
21notbid 285 . . 3  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( -.  ps 
<->  -.  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
32imbi1d 308 . 2  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ( -.  ps  ->  -.  ph )  <->  ( -.  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) )  ->  -.  ph ) ) )
41imbi2d 307 . . . 4  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ( ph  ->  ps )  <->  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) ) ) )
5 id 19 . . . . 5  |-  ( (
ph 
<->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )  ->  ( ph  <->  ( ( ps  /\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
65imbi2d 307 . . . 4  |-  ( (
ph 
<->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )  ->  ( ( ph  ->  ph )  <->  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) ) ) )
7 id 19 . . . 4  |-  ( ph  ->  ph )
84, 6, 7elimh 922 . . 3  |-  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )
98con3i 127 . 2  |-  ( -.  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) )  ->  -.  ph )
103, 9dedt 923 1  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360
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