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Theorem bj-bibibi 31174
Description: A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-bibibi  |-  ( ph  <->  ( ps  <->  ( ph  <->  ps )
) )

Proof of Theorem bj-bibibi
StepHypRef Expression
1 pm5.501 342 . 2  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
2 bianir 975 . . . 4  |-  ( ( ps  /\  ( ph  <->  ps ) )  ->  ph )
32ex 435 . . 3  |-  ( ps 
->  ( ( ph  <->  ps )  ->  ph ) )
4 bibif 347 . . . . 5  |-  ( -. 
ps  ->  ( ( ph  <->  ps )  <->  -.  ph ) )
54con2bid 330 . . . 4  |-  ( -. 
ps  ->  ( ph  <->  -.  ( ph 
<->  ps ) ) )
65biimprd 226 . . 3  |-  ( -. 
ps  ->  ( -.  ( ph 
<->  ps )  ->  ph )
)
73, 6bija 356 . 2  |-  ( ( ps  <->  ( ph  <->  ps )
)  ->  ph )
81, 7impbii 190 1  |-  ( ph  <->  ( ps  <->  ( ph  <->  ps )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-an 372
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator