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Theorem dedth4v 3612
Description: Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3610. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
Hypotheses
Ref Expression
dedth4v.1  |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ps  <->  ch )
)
dedth4v.2  |-  ( B  =  if ( ph ,  B ,  S )  ->  ( ch  <->  th )
)
dedth4v.3  |-  ( C  =  if ( ph ,  C ,  T )  ->  ( th  <->  ta )
)
dedth4v.4  |-  ( D  =  if ( ph ,  D ,  U )  ->  ( ta  <->  et )
)
dedth4v.5  |-  et
Assertion
Ref Expression
dedth4v  |-  ( ph  ->  ps )

Proof of Theorem dedth4v
StepHypRef Expression
1 dedth4v.1 . . . 4  |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ps  <->  ch )
)
2 dedth4v.2 . . . 4  |-  ( B  =  if ( ph ,  B ,  S )  ->  ( ch  <->  th )
)
3 dedth4v.3 . . . 4  |-  ( C  =  if ( ph ,  C ,  T )  ->  ( th  <->  ta )
)
4 dedth4v.4 . . . 4  |-  ( D  =  if ( ph ,  D ,  U )  ->  ( ta  <->  et )
)
5 dedth4v.5 . . . 4  |-  et
61, 2, 3, 4, 5dedth4h 3609 . . 3  |-  ( ( ( ph  /\  ph )  /\  ( ph  /\  ph ) )  ->  ps )
76anidms 626 . 2  |-  ( (
ph  /\  ph )  ->  ps )
87anidms 626 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   ifcif 3565
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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