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Theorem bi3 180
Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
Assertion
Ref Expression
bi3  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  ( ph  <->  ps ) ) )

Proof of Theorem bi3
StepHypRef Expression
1 df-bi 178 . . 3  |-  -.  (
( ( ph  <->  ps )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
) )  ->  -.  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )
2 simprim 144 . . 3  |-  ( -.  ( ( ( ph  <->  ps )  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )  ->  -.  ( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )  -> 
( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )
31, 2ax-mp 8 . 2  |-  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  -> 
( ph  <->  ps ) )
43expi 143 1  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  ( ph  <->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177
This theorem is referenced by:  impbii  181  impbidd  182  dfbi1  185  bisym  282  eqsbc3rVD  28661  orbi1rVD  28669  3impexpVD  28677  3impexpbicomVD  28678  imbi12VD  28694  sbcim2gVD  28696  sb5ALTVD  28734
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 178
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