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Theorem cases2 980
Description: Case disjunction according to the value of  ph. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)
Assertion
Ref Expression
cases2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )

Proof of Theorem cases2
StepHypRef Expression
1 pm3.4 563 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ph  ->  ps ) )
2 pm2.24 112 . . . . 5  |-  ( ph  ->  ( -.  ph  ->  ch ) )
32adantr 466 . . . 4  |-  ( (
ph  /\  ps )  ->  ( -.  ph  ->  ch ) )
41, 3jca 534 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ph  ->  ps )  /\  ( -. 
ph  ->  ch ) ) )
5 pm2.21 111 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
65adantr 466 . . . 4  |-  ( ( -.  ph  /\  ch )  ->  ( ph  ->  ps ) )
7 pm3.4 563 . . . 4  |-  ( ( -.  ph  /\  ch )  ->  ( -.  ph  ->  ch ) )
86, 7jca 534 . . 3  |-  ( ( -.  ph  /\  ch )  ->  ( ( ph  ->  ps )  /\  ( -. 
ph  ->  ch ) ) )
94, 8jaoi 380 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  -> 
( ( ph  ->  ps )  /\  ( -. 
ph  ->  ch ) ) )
10 pm2.27 40 . . . . . 6  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
)
1110imdistani 694 . . . . 5  |-  ( (
ph  /\  ( ph  ->  ps ) )  -> 
( ph  /\  ps )
)
1211orcd 393 . . . 4  |-  ( (
ph  /\  ( ph  ->  ps ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
1312adantrr 721 . . 3  |-  ( (
ph  /\  ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
14 pm2.27 40 . . . . . 6  |-  ( -. 
ph  ->  ( ( -. 
ph  ->  ch )  ->  ch ) )
1514imdistani 694 . . . . 5  |-  ( ( -.  ph  /\  ( -.  ph  ->  ch )
)  ->  ( -.  ph 
/\  ch ) )
1615olcd 394 . . . 4  |-  ( ( -.  ph  /\  ( -.  ph  ->  ch )
)  ->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch ) ) )
1716adantrl 720 . . 3  |-  ( ( -.  ph  /\  (
( ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
1813, 17pm2.61ian 797 . 2  |-  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
199, 18impbii 190 1  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( -.  ph  ->  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:  dfifp2  1422  ifval  3945  ifpidg  35882  ifpim123g  35891
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