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Theorem keephyp 3993
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 3965 . 2  |-  ( ( ps  /\  ch )  ->  th )
61, 2, 5mp2an 670 1  |-  th
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   ifcif 3929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-if 3930
This theorem is referenced by:  keepel  3996  boxcutc  7505  fin23lem13  8703  abvtrivd  17684  znf1o  18763  zntoslem  18768  dscmet  21259  sqff1o  23654  lgsne0  23806  dchrisum0flblem1  23891  dchrisum0flblem2  23892
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