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Theorem bisym1 26174
Description: A symmetry with  <->.

See negsym1 26172 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

Assertion
Ref Expression
bisym1  |-  ( ( ps  <->  ( ps  <->  F.  )
)  ->  ( ps  <->  ph ) )

Proof of Theorem bisym1
StepHypRef Expression
1 nbfal 1335 . . 3  |-  ( -. 
ps 
<->  ( ps  <->  F.  )
)
21bibi2i 306 . 2  |-  ( ( ps  <->  -.  ps )  <->  ( ps  <->  ( ps  <->  F.  )
) )
3 pm5.19 351 . . 3  |-  -.  ( ps 
<->  -.  ps )
43pm2.21i 126 . 2  |-  ( ( ps  <->  -.  ps )  ->  ( ps  <->  ph ) )
52, 4sylbir 206 1  |-  ( ( ps  <->  ( ps  <->  F.  )
)  ->  ( ps  <->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    F. wfal 1327
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 179  df-tru 1329  df-fal 1330
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