MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm3.2an3 Structured version   Unicode version

Theorem pm3.2an3 1167
Description: Version of pm3.2 447 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
pm3.2an3  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  /\  ps  /\  ch ) ) ) )

Proof of Theorem pm3.2an3
StepHypRef Expression
1 pm3.2 447 . . 3  |-  ( (
ph  /\  ps )  ->  ( ch  ->  (
( ph  /\  ps )  /\  ch ) ) )
21ex 434 . 2  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ( ph  /\ 
ps )  /\  ch ) ) ) )
3 df-3an 967 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
43bicomi 202 . 2  |-  ( ( ( ph  /\  ps )  /\  ch )  <->  ( ph  /\ 
ps  /\  ch )
)
52, 4syl8ib 231 1  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  /\  ps  /\  ch ) ) ) )
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:  3exp  1187  tratrb  31524  19.21a3con13vVD  31870  tratrbVD  31879
  Copyright terms: Public domain W3C validator