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Theorem dfbi1 186
Description: Relate the biconditional connective to primitive connectives. See dfbi1gb 187 for an unusual version proved directly from axioms. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
dfbi1  |-  ( (
ph 
<->  ps )  <->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )

Proof of Theorem dfbi1
StepHypRef Expression
1 df-bi 179 . . 3  |-  -.  (
( ( ph  <->  ps )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
) )  ->  -.  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )
2 simplim 145 . . 3  |-  ( -.  ( ( ( ph  <->  ps )  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )  ->  -.  ( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )  -> 
( ( ph  <->  ps )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
) ) )
31, 2ax-mp 10 . 2  |-  ( (
ph 
<->  ps )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
4 bi3 181 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  ( ph  <->  ps ) ) )
54impi 142 . 2  |-  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  -> 
( ph  <->  ps ) )
63, 5impbii 182 1  |-  ( (
ph 
<->  ps )  <->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178
This theorem is referenced by:  bi2  191  dfbi2  612  tbw-bijust  1458  rb-bijust  1509  nfbid  1728  axrepprim  23406  axacprim  23411
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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