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Theorem keephyp 3785
Description: Transform a hypothesis  ps that we want to keep (but contains the same class variable  A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
keephyp.1  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th )
)
keephyp.2  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch  <->  th )
)
keephyp.3  |-  ps
keephyp.4  |-  ch
Assertion
Ref Expression
keephyp  |-  th

Proof of Theorem keephyp
StepHypRef Expression
1 keephyp.3 . 2  |-  ps
2 keephyp.4 . 2  |-  ch
3 keephyp.1 . . 3  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th )
)
4 keephyp.2 . . 3  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch  <->  th )
)
53, 4ifboth 3762 . 2  |-  ( ( ps  /\  ch )  ->  th )
61, 2, 5mp2an 654 1  |-  th
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   ifcif 3731
This theorem is referenced by:  keepel  3788  boxcutc  7097  fin23lem13  8204  abvtrivd  15920  znf1o  16824  zntoslem  16829  dscmet  18612  sqff1o  20957  lgsne0  21109  dchrisum0flblem1  21194  dchrisum0flblem2  21195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-if 3732
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