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Theorem keephyp 3738
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 3715 . 2  |-  ( ( ps  /\  ch )  ->  th )
61, 2, 5mp2an 654 1  |-  th
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   ifcif 3684
This theorem is referenced by:  keepel  3741  boxcutc  7043  fin23lem13  8147  abvtrivd  15857  znf1o  16757  zntoslem  16762  dscmet  18493  sqff1o  20834  lgsne0  20986  dchrisum0flblem1  21071  dchrisum0flblem2  21072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-if 3685
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