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Theorem elimh 922
Description: Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page. (Contributed by NM, 26-Jun-2002.)
Hypotheses
Ref Expression
elimh.1  |-  ( (
ph 
<->  ( ( ph  /\  ch )  \/  ( ps  /\  -.  ch )
) )  ->  ( ch 
<->  ta ) )
elimh.2  |-  ( ( ps  <->  ( ( ph  /\ 
ch )  \/  ( ps  /\  -.  ch )
) )  ->  ( th 
<->  ta ) )
elimh.3  |-  th
Assertion
Ref Expression
elimh  |-  ta

Proof of Theorem elimh
StepHypRef Expression
1 dedlema 920 . . . 4  |-  ( ch 
->  ( ph  <->  ( ( ph  /\  ch )  \/  ( ps  /\  -.  ch ) ) ) )
2 elimh.1 . . . 4  |-  ( (
ph 
<->  ( ( ph  /\  ch )  \/  ( ps  /\  -.  ch )
) )  ->  ( ch 
<->  ta ) )
31, 2syl 15 . . 3  |-  ( ch 
->  ( ch  <->  ta )
)
43ibi 232 . 2  |-  ( ch 
->  ta )
5 elimh.3 . . 3  |-  th
6 dedlemb 921 . . . 4  |-  ( -. 
ch  ->  ( ps  <->  ( ( ph  /\  ch )  \/  ( ps  /\  -.  ch ) ) ) )
7 elimh.2 . . . 4  |-  ( ( ps  <->  ( ( ph  /\ 
ch )  \/  ( ps  /\  -.  ch )
) )  ->  ( th 
<->  ta ) )
86, 7syl 15 . . 3  |-  ( -. 
ch  ->  ( th  <->  ta )
)
95, 8mpbii 202 . 2  |-  ( -. 
ch  ->  ta )
104, 9pm2.61i 156 1  |-  ta
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
This theorem is referenced by:  con3th  924
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
  Copyright terms: Public domain W3C validator