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

Theorem 3anim123d 1296
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 563 . . 3  |-  ( ph  ->  ( ( ps  /\  th )  ->  ( ch  /\ 
ta ) ) )
4 3anim123d.3 . . 3  |-  ( ph  ->  ( et  ->  ze )
)
53, 4anim12d 563 . 2  |-  ( ph  ->  ( ( ( ps 
/\  th )  /\  et )  ->  ( ( ch 
/\  ta )  /\  ze ) ) )
6 df-3an 967 . 2  |-  ( ( ps  /\  th  /\  et )  <->  ( ( ps 
/\  th )  /\  et ) )
7 df-3an 967 . 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 965
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 967
This theorem is referenced by:  pofun  4662  isopolem  6041  issmo2  6815  smores  6818  inawina  8862  gchina  8871  repswcshw  12451  issubmnd  15454  issubg2  15701  issubrg2  16890  ocv2ss  18103  sslm  18908  cmetcaulem  20804  axcontlem4  23218  axcontlem8  23222  redwlk  23510  3cycl3dv  23533  3v3e3cycl1  23535  constr3trllem5  23545  grponnncan2  23746  dipsubdir  24253  cgr3tr4  28088  idinside  28120  ftc1anclem7  28478  fzmul  28641  fdc1  28647  rngosubdi  28764  rngosubdir  28765  el2wlkonotot0  30396  cdlemg33a  34355
  Copyright terms: Public domain W3C validator