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Theorem bi3 181
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 179 . . 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 10 . 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 5    -> wi 6    <-> wb 178
This theorem is referenced by:  impbii  182  impbidd  183  dfbi1  186  bisym  283  eqsbc3rVD  27884  orbi1rVD  27892  3impexpVD  27900  3impexpbicomVD  27901  imbi12VD  27917  sbcim2gVD  27919  sb5ALTVD  27957
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179
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