MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dedlem0b Unicode version

Theorem dedlem0b 919
Description: Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
Assertion
Ref Expression
dedlem0b  |-  ( -. 
ph  ->  ( ps  <->  ( ( ps  ->  ph )  ->  ( ch  /\  ph ) ) ) )

Proof of Theorem dedlem0b
StepHypRef Expression
1 pm2.21 100 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  ( ch  /\  ph )
) )
21imim2d 48 . . 3  |-  ( -. 
ph  ->  ( ( ps 
->  ph )  ->  ( ps  ->  ( ch  /\  ph ) ) ) )
32com23 72 . 2  |-  ( -. 
ph  ->  ( ps  ->  ( ( ps  ->  ph )  ->  ( ch  /\  ph ) ) ) )
4 pm2.21 100 . . . . 5  |-  ( -. 
ps  ->  ( ps  ->  ph ) )
5 simpr 447 . . . . 5  |-  ( ( ch  /\  ph )  ->  ph )
64, 5imim12i 53 . . . 4  |-  ( ( ( ps  ->  ph )  ->  ( ch  /\  ph ) )  ->  ( -.  ps  ->  ph ) )
76con1d 116 . . 3  |-  ( ( ( ps  ->  ph )  ->  ( ch  /\  ph ) )  ->  ( -.  ph  ->  ps )
)
87com12 27 . 2  |-  ( -. 
ph  ->  ( ( ( ps  ->  ph )  -> 
( ch  /\  ph ) )  ->  ps ) )
93, 8impbid 183 1  |-  ( -. 
ph  ->  ( ps  <->  ( ( ps  ->  ph )  ->  ( ch  /\  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ 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-an 360
  Copyright terms: Public domain W3C validator