MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3anim123d Structured version   Unicode version

Theorem 3anim123d 1291
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 560 . . 3  |-  ( ph  ->  ( ( ps  /\  th )  ->  ( ch  /\ 
ta ) ) )
4 3anim123d.3 . . 3  |-  ( ph  ->  ( et  ->  ze )
)
53, 4anim12d 560 . 2  |-  ( ph  ->  ( ( ( ps 
/\  th )  /\  et )  ->  ( ( ch 
/\  ta )  /\  ze ) ) )
6 df-3an 962 . 2  |-  ( ( ps  /\  th  /\  et )  <->  ( ( ps 
/\  th )  /\  et ) )
7 df-3an 962 . 2  |-  ( ( ch  /\  ta  /\  ze )  <->  ( ( ch 
/\  ta )  /\  ze ) )
85, 6, 73imtr4g 270 1  |-  ( ph  ->  ( ( ps  /\  th 
/\  et )  -> 
( ch  /\  ta  /\ 
ze ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960
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 185  df-an 371  df-3an 962
This theorem is referenced by:  pofun  4653  isopolem  6033  issmo2  6806  smores  6809  inawina  8853  gchina  8862  repswcshw  12442  issubmnd  15445  issubg2  15689  issubrg2  16865  ocv2ss  18057  sslm  18862  cmetcaulem  20758  axcontlem4  23148  axcontlem8  23152  redwlk  23440  3cycl3dv  23463  3v3e3cycl1  23465  constr3trllem5  23475  grponnncan2  23676  dipsubdir  24183  cgr3tr4  28012  idinside  28044  ftc1anclem7  28398  fzmul  28561  fdc1  28567  rngosubdi  28684  rngosubdir  28685  el2wlkonotot0  30316  cdlemg33a  34072
  Copyright terms: Public domain W3C validator