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Theorem and4as 24939
Description: A consequence of  /\ associativity in a triple conjunct. (Contributed by FL, 14-Jul-2007.)
Assertion
Ref Expression
and4as  |-  ( (
ph  /\  ps  /\  ( ch  /\  th ) )  <-> 
( ( ph  /\  ps  /\  ch )  /\  th ) )

Proof of Theorem and4as
StepHypRef Expression
1 pm3.2 434 . . . . 5  |-  ( (
ph  /\  ps  /\  ch )  ->  ( th  ->  ( ( ph  /\  ps  /\ 
ch )  /\  th ) ) )
213exp 1150 . . . 4  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ( ph  /\  ps  /\ 
ch )  /\  th ) ) ) ) )
32imp4a 572 . . 3  |-  ( ph  ->  ( ps  ->  (
( ch  /\  th )  ->  ( ( ph  /\ 
ps  /\  ch )  /\  th ) ) ) )
433imp 1145 . 2  |-  ( (
ph  /\  ps  /\  ( ch  /\  th ) )  ->  ( ( ph  /\ 
ps  /\  ch )  /\  th ) )
5 id 19 . . . . 5  |-  ( (
ph  /\  ps  /\  ( ch  /\  th ) )  ->  ( ph  /\  ps  /\  ( ch  /\  th ) ) )
653exp 1150 . . . 4  |-  ( ph  ->  ( ps  ->  (
( ch  /\  th )  ->  ( ph  /\  ps  /\  ( ch  /\  th ) ) ) ) )
76exp4a 589 . . 3  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  (
ph  /\  ps  /\  ( ch  /\  th ) ) ) ) ) )
873imp1 1164 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  th )  ->  ( ph  /\  ps  /\  ( ch  /\  th ) ) )
94, 8impbii 180 1  |-  ( (
ph  /\  ps  /\  ( ch  /\  th ) )  <-> 
( ( ph  /\  ps  /\  ch )  /\  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934
This theorem is referenced by:  and4com  24940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
  Copyright terms: Public domain W3C validator