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Theorem con3th 929
Description: Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 128 demonstrates the use of the weak deduction theorem dedt 928 to derive it from con3i 129. (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 21 . . . 4  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ps  <->  ( ( ps  /\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
21notbid 287 . . 3  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( -.  ps 
<->  -.  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
32imbi1d 310 . 2  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ( -.  ps  ->  -.  ph )  <->  ( -.  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) )  ->  -.  ph ) ) )
41imbi2d 309 . . . 4  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ( ph  ->  ps )  <->  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) ) ) )
5 id 21 . . . . 5  |-  ( (
ph 
<->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )  ->  ( ph  <->  ( ( ps  /\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
65imbi2d 309 . . . 4  |-  ( (
ph 
<->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )  ->  ( ( ph  ->  ph )  <->  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) ) ) )
7 id 21 . . . 4  |-  ( ph  ->  ph )
84, 6, 7elimh 927 . . 3  |-  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )
98con3i 129 . 2  |-  ( -.  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) )  ->  -.  ph )
103, 9dedt 928 1  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362
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