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Theorem dedth3v 3996
Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3995. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
Hypotheses
Ref Expression
dedth3v.1  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ps  <->  ch )
)
dedth3v.2  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
dedth3v.3  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
dedth3v.4  |-  ta
Assertion
Ref Expression
dedth3v  |-  ( ph  ->  ps )

Proof of Theorem dedth3v
StepHypRef Expression
1 dedth3v.1 . . . 4  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ps  <->  ch )
)
2 dedth3v.2 . . . 4  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
3 dedth3v.3 . . . 4  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
4 dedth3v.4 . . . 4  |-  ta
51, 2, 3, 4dedth3h 3993 . . 3  |-  ( (
ph  /\  ph  /\  ph )  ->  ps )
653anidm12 1285 . 2  |-  ( (
ph  /\  ph )  ->  ps )
76anidms 645 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   ifcif 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-if 3940
This theorem is referenced by:  sseliALT  4578
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