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Theorem cases2OLD 972
Description: Obsolete proof of cases2 971 as of 2-Jan-2020. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cases2OLD  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )

Proof of Theorem cases2OLD
StepHypRef Expression
1 pm3.4 561 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ph  ->  ps ) )
2 simpl 457 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
3 pm2.24 109 . . . . 5  |-  ( ph  ->  ( -.  ph  ->  ch ) )
42, 3syl 16 . . . 4  |-  ( (
ph  /\  ps )  ->  ( -.  ph  ->  ch ) )
51, 4jca 532 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ph  ->  ps )  /\  ( -. 
ph  ->  ch ) ) )
6 simpl 457 . . . . 5  |-  ( ( -.  ph  /\  ch )  ->  -.  ph )
7 pm2.21 108 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
86, 7syl 16 . . . 4  |-  ( ( -.  ph  /\  ch )  ->  ( ph  ->  ps ) )
9 pm3.4 561 . . . 4  |-  ( ( -.  ph  /\  ch )  ->  ( -.  ph  ->  ch ) )
108, 9jca 532 . . 3  |-  ( ( -.  ph  /\  ch )  ->  ( ( ph  ->  ps )  /\  ( -. 
ph  ->  ch ) ) )
115, 10jaoi 379 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  -> 
( ( ph  ->  ps )  /\  ( -. 
ph  ->  ch ) ) )
12 exmid 415 . . 3  |-  ( ph  \/  -.  ph )
13 pm2.27 39 . . . . . . 7  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
)
1413imdistani 690 . . . . . 6  |-  ( (
ph  /\  ( ph  ->  ps ) )  -> 
( ph  /\  ps )
)
1514orcd 392 . . . . 5  |-  ( (
ph  /\  ( ph  ->  ps ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
1615ex 434 . . . 4  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch ) ) ) )
17 pm2.27 39 . . . . . . 7  |-  ( -. 
ph  ->  ( ( -. 
ph  ->  ch )  ->  ch ) )
1817imdistani 690 . . . . . 6  |-  ( ( -.  ph  /\  ( -.  ph  ->  ch )
)  ->  ( -.  ph 
/\  ch ) )
1918olcd 393 . . . . 5  |-  ( ( -.  ph  /\  ( -.  ph  ->  ch )
)  ->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch ) ) )
2019ex 434 . . . 4  |-  ( -. 
ph  ->  ( ( -. 
ph  ->  ch )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) ) )
2116, 20jaoa 510 . . 3  |-  ( (
ph  \/  -.  ph )  ->  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch )
)  ->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch ) ) ) )
2212, 21ax-mp 5 . 2  |-  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
2311, 22impbii 188 1  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369
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 185  df-or 370  df-an 371
This theorem is referenced by: (None)
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