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Theorem boxand 25006
Description: Distributivity of  [.] over  /\. (Contributed by FL, 1-Sep-2016.)
Assertion
Ref Expression
boxand  |-  ( [.] ( ph  /\  ps ) 
<->  ( [.] ph  /\  [.] ps ) )

Proof of Theorem boxand
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( (
ph  /\  ps )  ->  ph )
21impbox 24981 . . 3  |-  ( [.] ( ph  /\  ps )  ->  [.] ph )
3 simpr 447 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
43impbox 24981 . . 3  |-  ( [.] ( ph  /\  ps )  ->  [.] ps )
52, 4jca 518 . 2  |-  ( [.] ( ph  /\  ps )  ->  ( [.] ph  /\ 
[.] ps ) )
6 pm3.2 434 . . . 4  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
76impbox2 25005 . . 3  |-  ( [.] ph  ->  ( [.] ps  ->  [.] ( ph  /\  ps ) ) )
87imp 418 . 2  |-  ( ( [.] ph  /\  [.] ps )  ->  [.] ( ph  /\  ps ) )
95, 8impbii 180 1  |-  ( [.] ( ph  /\  ps ) 
<->  ( [.] ph  /\  [.] ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   [.]wbox 24970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl1 24974  ax-lmp 24978
This theorem depends on definitions:  df-bi 177  df-an 360
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