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Theorem an3andi 1377
Description: Distribution of conjunction over conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
Assertion
Ref Expression
an3andi  |-  ( (
ph  /\  ( ps  /\ 
ch  /\  th )
)  <->  ( ( ph  /\ 
ps )  /\  ( ph  /\  ch )  /\  ( ph  /\  th )
) )

Proof of Theorem an3andi
StepHypRef Expression
1 df-3an 984 . . . 4  |-  ( ( ps  /\  ch  /\  th )  <->  ( ( ps 
/\  ch )  /\  th ) )
21anbi2i 698 . . 3  |-  ( (
ph  /\  ( ps  /\ 
ch  /\  th )
)  <->  ( ph  /\  ( ( ps  /\  ch )  /\  th )
) )
3 anandi 835 . . 3  |-  ( (
ph  /\  ( ( ps  /\  ch )  /\  th ) )  <->  ( ( ph  /\  ( ps  /\  ch ) )  /\  ( ph  /\  th ) ) )
4 anandi 835 . . . 4  |-  ( (
ph  /\  ( ps  /\ 
ch ) )  <->  ( ( ph  /\  ps )  /\  ( ph  /\  ch )
) )
54anbi1i 699 . . 3  |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  ( ph  /\  th ) )  <->  ( (
( ph  /\  ps )  /\  ( ph  /\  ch ) )  /\  ( ph  /\  th ) ) )
62, 3, 53bitri 274 . 2  |-  ( (
ph  /\  ( ps  /\ 
ch  /\  th )
)  <->  ( ( (
ph  /\  ps )  /\  ( ph  /\  ch ) )  /\  ( ph  /\  th ) ) )
7 df-3an 984 . 2  |-  ( ( ( ph  /\  ps )  /\  ( ph  /\  ch )  /\  ( ph  /\  th ) )  <-> 
( ( ( ph  /\ 
ps )  /\  ( ph  /\  ch ) )  /\  ( ph  /\  th ) ) )
86, 7bitr4i 255 1  |-  ( (
ph  /\  ( ps  /\ 
ch  /\  th )
)  <->  ( ( ph  /\ 
ps )  /\  ( ph  /\  ch )  /\  ( ph  /\  th )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982
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 188  df-an 372  df-3an 984
This theorem is referenced by:  raltpd  4123
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