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Theorem keephyp 3619
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 3596 . 2  |-  ( ( ps  /\  ch )  ->  th )
61, 2, 5mp2an 653 1  |-  th
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   ifcif 3565
This theorem is referenced by:  keepel  3622  boxcutc  6859  fin23lem13  7958  abvtrivd  15605  znf1o  16505  zntoslem  16510  dscmet  18095  sqff1o  20420  lgsne0  20572  dchrisum0flblem1  20657  dchrisum0flblem2  20658
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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