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Theorem biimpexp 24070
Description: A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
Assertion
Ref Expression
biimpexp  |-  ( ( ( ph  <->  ps )  ->  ch )  <->  ( ( ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  ch )
) )

Proof of Theorem biimpexp
StepHypRef Expression
1 dfbi2 609 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21imbi1i 315 . 2  |-  ( ( ( ph  <->  ps )  ->  ch )  <->  ( (
( ph  ->  ps )  /\  ( ps  ->  ph )
)  ->  ch )
)
3 impexp 433 . 2  |-  ( ( ( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  ->  ch )  <->  ( ( ph  ->  ps )  ->  (
( ps  ->  ph )  ->  ch ) ) )
42, 3bitri 240 1  |-  ( ( ( ph  <->  ps )  ->  ch )  <->  ( ( ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  axextdfeq  24154
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
  Copyright terms: Public domain W3C validator