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Theorem dfbi1 194
Description: Relate the biconditional connective to primitive connectives. See dfbi1ALT 195 for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.)
Assertion
Ref Expression
dfbi1  |-  ( (
ph 
<->  ps )  <->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )

Proof of Theorem dfbi1
StepHypRef Expression
1 df-bi 188 . . 3  |-  -.  (
( ( ph  <->  ps )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
) )  ->  -.  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )
2 simplim 154 . . 3  |-  ( -.  ( ( ( ph  <->  ps )  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )  ->  -.  ( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  ( ph  <->  ps ) ) )  -> 
( ( ph  <->  ps )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph )
) ) )
31, 2ax-mp 5 . 2  |-  ( (
ph 
<->  ps )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
4 impbi 189 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  ( ph  <->  ps ) ) )
54impi 151 . 2  |-  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  -> 
( ph  <->  ps ) )
63, 5impbii 190 1  |-  ( (
ph 
<->  ps )  <->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187
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 188
This theorem is referenced by:  biimpr  201  dfbi2  632  tbw-bijust  1577  rb-bijust  1628  axrepprim  30114  axacprim  30119
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