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Theorem dfbi 610
Description: Definition df-bi 177 rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008.)
Assertion
Ref Expression
dfbi  |-  ( ( ( ph  <->  ps )  ->  ( ( ph  ->  ps )  /\  ( ps 
->  ph ) ) )  /\  ( ( (
ph  ->  ps )  /\  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )

Proof of Theorem dfbi
StepHypRef Expression
1 dfbi2 609 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21biimpi 186 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ph  ->  ps )  /\  ( ps  ->  ph )
) )
31biimpri 197 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  ( ph 
<->  ps ) )
42, 3pm3.2i 441 1  |-  ( ( ( ph  <->  ps )  ->  ( ( ph  ->  ps )  /\  ( ps 
->  ph ) ) )  /\  ( ( (
ph  ->  ps )  /\  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
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
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