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Theorem 3anim123d 1262
Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
3anim123d.1  |-  ( ph  ->  ( ps  ->  ch ) )
3anim123d.2  |-  ( ph  ->  ( th  ->  ta ) )
3anim123d.3  |-  ( ph  ->  ( et  ->  ze )
)
Assertion
Ref Expression
3anim123d  |-  ( ph  ->  ( ( ps  /\  th 
/\  et )  -> 
( ch  /\  ta  /\ 
ze ) ) )

Proof of Theorem 3anim123d
StepHypRef Expression
1 3anim123d.1 . . . 4  |-  ( ph  ->  ( ps  ->  ch ) )
2 3anim123d.2 . . . 4  |-  ( ph  ->  ( th  ->  ta ) )
31, 2anim12d 548 . . 3  |-  ( ph  ->  ( ( ps  /\  th )  ->  ( ch  /\ 
ta ) ) )
4 3anim123d.3 . . 3  |-  ( ph  ->  ( et  ->  ze )
)
53, 4anim12d 548 . 2  |-  ( ph  ->  ( ( ( ps 
/\  th )  /\  et )  ->  ( ( ch 
/\  ta )  /\  ze ) ) )
6 df-3an 939 . 2  |-  ( ( ps  /\  th  /\  et )  <->  ( ( ps 
/\  th )  /\  et ) )
7 df-3an 939 . 2  |-  ( ( ch  /\  ta  /\  ze )  <->  ( ( ch 
/\  ta )  /\  ze ) )
85, 6, 73imtr4g 263 1  |-  ( ph  ->  ( ( ps  /\  th 
/\  et )  -> 
( ch  /\  ta  /\ 
ze ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937
This theorem is referenced by:  pofun  4548  isopolem  6094  issmo2  6640  smores  6643  inawina  8596  gchina  8605  issubmnd  14755  issubg2  14990  issubrg2  15919  ocv2ss  16931  sslm  17394  cmetcaulem  19272  redwlk  21637  3cycl3dv  21660  3v3e3cycl1  21662  constr3trllem5  21672  grponnncan2  21873  dipsubdir  22380  axcontlem4  25937  axcontlem8  25941  cgr3tr4  26017  idinside  26049  ftc1anclem7  26324  fzmul  26482  fdc1  26488  rngosubdi  26607  rngosubdir  26608  el2wlkonotot0  28428  cdlemg33a  31601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939
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