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Theorem bj-andnotim 31177
Description: Two ways of expressing a certain ternary connective. Note the respective positions of the three wff's. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-andnotim  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  <->  ( ( ph  ->  ps )  \/  ch )
)

Proof of Theorem bj-andnotim
StepHypRef Expression
1 imor 413 . . 3  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  <->  ( -.  ( ph  /\  -.  ps )  \/  ch ) )
2 iman 425 . . . . 5  |-  ( (
ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
)
32biimpri 209 . . . 4  |-  ( -.  ( ph  /\  -.  ps )  ->  ( ph  ->  ps ) )
43orim1i 519 . . 3  |-  ( ( -.  ( ph  /\  -.  ps )  \/  ch )  ->  ( ( ph  ->  ps )  \/  ch ) )
51, 4sylbi 198 . 2  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  ->  ( ( ph  ->  ps )  \/  ch )
)
6 pm2.24 112 . . . . 5  |-  ( ps 
->  ( -.  ps  ->  ch ) )
76imim2i 16 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ( -. 
ps  ->  ch ) ) )
87impd 432 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ph  /\  -.  ps )  ->  ch ) )
9 ax-1 6 . . 3  |-  ( ch 
->  ( ( ph  /\  -.  ps )  ->  ch ) )
108, 9jaoi 380 . 2  |-  ( ( ( ph  ->  ps )  \/  ch )  ->  ( ( ph  /\  -.  ps )  ->  ch ) )
115, 10impbii 190 1  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  <->  ( ( ph  ->  ps )  \/  ch )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370
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  df-or 371  df-an 372
This theorem is referenced by: (None)
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