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Theorem elimdhyp 3618
Description: Version of elimhyp 3613 where the hypothesis is deduced from the final antecedent. See ghomgrplem 23996 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 3571 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
32eqcomd 2288 . . . 4  |-  ( ph  ->  A  =  if (
ph ,  A ,  B ) )
4 elimdhyp.2 . . . 4  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch )
)
53, 4syl 15 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
61, 5mpbid 201 . 2  |-  ( ph  ->  ch )
7 elimdhyp.4 . . 3  |-  th
8 iffalse 3572 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
98eqcomd 2288 . . . 4  |-  ( -. 
ph  ->  B  =  if ( ph ,  A ,  B ) )
10 elimdhyp.3 . . . 4  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( th  <->  ch )
)
119, 10syl 15 . . 3  |-  ( -. 
ph  ->  ( th  <->  ch )
)
127, 11mpbii 202 . 2  |-  ( -. 
ph  ->  ch )
136, 12pm2.61i 156 1  |-  ch
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623   ifcif 3565
This theorem is referenced by:  divalg  12602  ghomgrplem  23996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-if 3566
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