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Theorem dedth4h 3911
Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3909. (Contributed by NM, 16-May-1999.)
Hypotheses
Ref Expression
dedth4h.1  |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ta  <->  et )
)
dedth4h.2  |-  ( B  =  if ( ps ,  B ,  S
)  ->  ( et  <->  ze ) )
dedth4h.3  |-  ( C  =  if ( ch ,  C ,  F
)  ->  ( ze  <->  si ) )
dedth4h.4  |-  ( D  =  if ( th ,  D ,  G
)  ->  ( si  <->  rh ) )
dedth4h.5  |-  rh
Assertion
Ref Expression
dedth4h  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )

Proof of Theorem dedth4h
StepHypRef Expression
1 dedth4h.1 . . . 4  |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ta  <->  et )
)
21imbi2d 314 . . 3  |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ( ( ch  /\  th )  ->  ta )  <->  ( ( ch  /\  th )  ->  et ) ) )
3 dedth4h.2 . . . 4  |-  ( B  =  if ( ps ,  B ,  S
)  ->  ( et  <->  ze ) )
43imbi2d 314 . . 3  |-  ( B  =  if ( ps ,  B ,  S
)  ->  ( (
( ch  /\  th )  ->  et )  <->  ( ( ch  /\  th )  ->  ze ) ) )
5 dedth4h.3 . . . 4  |-  ( C  =  if ( ch ,  C ,  F
)  ->  ( ze  <->  si ) )
6 dedth4h.4 . . . 4  |-  ( D  =  if ( th ,  D ,  G
)  ->  ( si  <->  rh ) )
7 dedth4h.5 . . . 4  |-  rh
85, 6, 7dedth2h 3909 . . 3  |-  ( ( ch  /\  th )  ->  ze )
92, 4, 8dedth2h 3909 . 2  |-  ( (
ph  /\  ps )  ->  ( ( ch  /\  th )  ->  ta )
)
109imp 427 1  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399   ifcif 3857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-if 3858
This theorem is referenced by:  dedth4v  3914  fprg  5982  omopth  7225  nn0opth2  12254  ax5seglem8  24360  hvsubsub4  26094  norm3lemt  26186  eigorth  26873
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