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Theorem con3th 949
Description: Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 134 demonstrates the use of the weak deduction theorem dedt 948 to derive it from con3i 135. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
con3th  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )

Proof of Theorem con3th
StepHypRef Expression
1 id 22 . . . 4  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ps  <->  ( ( ps  /\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
21notbid 294 . . 3  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( -.  ps 
<->  -.  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
32imbi1d 317 . 2  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ( -.  ps  ->  -.  ph )  <->  ( -.  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) )  ->  -.  ph ) ) )
41imbi2d 316 . . . 4  |-  ( ( ps  <->  ( ( ps 
/\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) )  ->  ( ( ph  ->  ps )  <->  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) ) ) )
5 id 22 . . . . 5  |-  ( (
ph 
<->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )  ->  ( ph  <->  ( ( ps  /\  ( ph  ->  ps ) )  \/  ( ph  /\  -.  ( ph  ->  ps ) ) ) ) )
65imbi2d 316 . . . 4  |-  ( (
ph 
<->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )  ->  ( ( ph  ->  ph )  <->  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) ) ) )
7 id 22 . . . 4  |-  ( ph  ->  ph )
84, 6, 7elimh 947 . . 3  |-  ( ph  ->  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) ) )
98con3i 135 . 2  |-  ( -.  ( ( ps  /\  ( ph  ->  ps )
)  \/  ( ph  /\ 
-.  ( ph  ->  ps ) ) )  ->  -.  ph )
103, 9dedt 948 1  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369
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  df-or 370  df-an 371
This theorem is referenced by: (None)
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