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Theorem aibandbiaiffaiffb 27965
Description: A closed form showing (a implies b and b implies a) same-as (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)
Assertion
Ref Expression
aibandbiaiffaiffb  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( ph  <->  ps ) )

Proof of Theorem aibandbiaiffaiffb
StepHypRef Expression
1 dfbi2 609 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
2 bicom 191 . . 3  |-  ( ( ( ph  <->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )  <->  ( (
( ph  ->  ps )  /\  ( ps  ->  ph )
)  <->  ( ph  <->  ps )
) )
32biimpi 186 . 2  |-  ( ( ( ph  <->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )  -> 
( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( ph  <->  ps ) ) )
41, 3ax-mp 8 1  |-  ( ( ( 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
  Copyright terms: Public domain W3C validator