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Theorem elimdhyp 3850
Description: Version of elimhyp 3845 where the hypothesis is deduced from the final antecedent. See ghomgrplem 27237 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
Hypotheses
Ref Expression
elimdhyp.1  |-  ( ph  ->  ps )
elimdhyp.2  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch )
)
elimdhyp.3  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( th  <->  ch )
)
elimdhyp.4  |-  th
Assertion
Ref Expression
elimdhyp  |-  ch

Proof of Theorem elimdhyp
StepHypRef Expression
1 elimdhyp.1 . . 3  |-  ( ph  ->  ps )
2 iftrue 3794 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
32eqcomd 2446 . . . 4  |-  ( ph  ->  A  =  if (
ph ,  A ,  B ) )
4 elimdhyp.2 . . . 4  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch )
)
53, 4syl 16 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
61, 5mpbid 210 . 2  |-  ( ph  ->  ch )
7 elimdhyp.4 . . 3  |-  th
8 iffalse 3796 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
98eqcomd 2446 . . . 4  |-  ( -. 
ph  ->  B  =  if ( ph ,  A ,  B ) )
10 elimdhyp.3 . . . 4  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( th  <->  ch )
)
119, 10syl 16 . . 3  |-  ( -. 
ph  ->  ( th  <->  ch )
)
127, 11mpbii 211 . 2  |-  ( -. 
ph  ->  ch )
136, 12pm2.61i 164 1  |-  ch
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1364   ifcif 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-if 3789
This theorem is referenced by:  divalg  13603  ghomgrplem  27237
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