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Theorem dedth3h 3811
Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3810. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
dedth3h.1  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( th  <->  ta )
)
dedth3h.2  |-  ( B  =  if ( ps ,  B ,  R
)  ->  ( ta  <->  et ) )
dedth3h.3  |-  ( C  =  if ( ch ,  C ,  S
)  ->  ( et  <->  ze ) )
dedth3h.4  |-  ze
Assertion
Ref Expression
dedth3h  |-  ( (
ph  /\  ps  /\  ch )  ->  th )

Proof of Theorem dedth3h
StepHypRef Expression
1 dedth3h.1 . . . 4  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( th  <->  ta )
)
21imbi2d 309 . . 3  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ( ( ps  /\  ch )  ->  th )  <->  ( ( ps  /\  ch )  ->  ta ) ) )
3 dedth3h.2 . . . 4  |-  ( B  =  if ( ps ,  B ,  R
)  ->  ( ta  <->  et ) )
4 dedth3h.3 . . . 4  |-  ( C  =  if ( ch ,  C ,  S
)  ->  ( et  <->  ze ) )
5 dedth3h.4 . . . 4  |-  ze
63, 4, 5dedth2h 3810 . . 3  |-  ( ( ps  /\  ch )  ->  ta )
72, 6dedth 3809 . 2  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
873impib 1152 1  |-  ( (
ph  /\  ps  /\  ch )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1654   ifcif 3767
This theorem is referenced by:  dedth3v  3814  faclbnd4lem2  11623  dvdsle  12933  gcdaddm  13067  ipdiri  22369  hvaddcan  22610  hvsubadd  22617  norm3dif  22690  omlsii  22943  chjass  23073  ledi  23080  spansncv  23193  pjcjt2  23232  pjopyth  23260  hoaddass  23323  hocsubdir  23326  hoddi  23531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-if 3768
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