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Theorem bi3 179
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 177 . . 3  |-  -.  (
( ( ph  <->  ps )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
) )  ->  -.  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )
2 simprim 142 . . 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 141 1  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  ( ph  <->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176
This theorem is referenced by:  impbii  180  impbidd  181  dfbi1  184  bisym  281  eqsbc3rVD  28932  orbi1rVD  28940  3impexpVD  28948  3impexpbicomVD  28949  imbi12VD  28965  sbcim2gVD  28967  sb5ALTVD  29005
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
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