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Theorem bibox 24982
Description: If  ph  <->  ps is unconditionally true then  ph is always true is equivalent to  ps is always true. (Contributed by FL, 22-Feb-2011.)
Hypothesis
Ref Expression
bibox.1  |-  ( ph  <->  ps )
Assertion
Ref Expression
bibox  |-  ( [.] ph 
<->  [.] ps )

Proof of Theorem bibox
StepHypRef Expression
1 bibox.1 . . . 4  |-  ( ph  <->  ps )
21biimpi 186 . . 3  |-  ( ph  ->  ps )
32impbox 24981 . 2  |-  ( [.] ph  ->  [.] ps )
41biimpri 197 . . 3  |-  ( ps 
->  ph )
54impbox 24981 . 2  |-  ( [.]
ps  ->  [.] ph )
63, 5impbii 180 1  |-  ( [.] ph 
<->  [.] ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [.]wbox 24970
This theorem is referenced by:  boxeq  24987  ltl4ev  24992  evevifev  25014
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
  Copyright terms: Public domain W3C validator