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Theorem dedth3h 3988
Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3987. (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 316 . . 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 3987 . . 3  |-  ( ( ps  /\  ch )  ->  ta )
72, 6dedth 3986 . 2  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
873impib 1189 1  |-  ( (
ph  /\  ps  /\  ch )  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   ifcif 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-if 3935
This theorem is referenced by:  dedth3v  3991  faclbnd4lem2  12329  dvdsle  13881  gcdaddm  14017  ipdiri  25409  hvaddcan  25651  hvsubadd  25658  norm3dif  25731  omlsii  25985  chjass  26115  ledi  26122  spansncv  26235  pjcjt2  26274  pjopyth  26302  hoaddass  26365  hocsubdir  26368  hoddi  26573
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