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Theorem ifpim23g 36109
Description: Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpim23g  |-  ( ( ( ph  ->  ps ) 
<-> if- ( ch ,  ps ,  -.  ph ) )  <-> 
( ( ( ph  /\ 
ps )  ->  ch )  /\  ( ch  ->  (
ph  \/  ps )
) ) )

Proof of Theorem ifpim23g
StepHypRef Expression
1 ifpidg 36105 . 2  |-  ( ( ( ph  ->  ps ) 
<-> if- ( ch ,  ps ,  -.  ph ) )  <-> 
( ( ( ( ch  /\  ps )  ->  ( ph  ->  ps ) )  /\  (
( ch  /\  ( ph  ->  ps ) )  ->  ps ) )  /\  ( ( -. 
ph  ->  ( ch  \/  ( ph  ->  ps )
) )  /\  (
( ph  ->  ps )  ->  ( ch  \/  -.  ph ) ) ) ) )
2 dfor2 412 . . . . 5  |-  ( (
ph  \/  ps )  <->  ( ( ph  ->  ps )  ->  ps ) )
32imbi2i 313 . . . 4  |-  ( ( ch  ->  ( ph  \/  ps ) )  <->  ( ch  ->  ( ( ph  ->  ps )  ->  ps )
) )
4 impexp 447 . . . 4  |-  ( ( ( ch  /\  ( ph  ->  ps ) )  ->  ps )  <->  ( ch  ->  ( ( ph  ->  ps )  ->  ps )
) )
5 ax-1 6 . . . . . 6  |-  ( ps 
->  ( ph  ->  ps ) )
65adantl 467 . . . . 5  |-  ( ( ch  /\  ps )  ->  ( ph  ->  ps ) )
76biantrur 508 . . . 4  |-  ( ( ( ch  /\  ( ph  ->  ps ) )  ->  ps )  <->  ( (
( ch  /\  ps )  ->  ( ph  ->  ps ) )  /\  (
( ch  /\  ( ph  ->  ps ) )  ->  ps ) ) )
83, 4, 73bitr2i 276 . . 3  |-  ( ( ch  ->  ( ph  \/  ps ) )  <->  ( (
( ch  /\  ps )  ->  ( ph  ->  ps ) )  /\  (
( ch  /\  ( ph  ->  ps ) )  ->  ps ) ) )
9 impexp 447 . . . . 5  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ph  ->  ( ps  ->  ch ) ) )
10 imdi 364 . . . . . 6  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ( ph  ->  ps )  -> 
( ph  ->  ch )
) )
11 imor 413 . . . . . . . 8  |-  ( (
ph  ->  ch )  <->  ( -.  ph  \/  ch ) )
12 orcom 388 . . . . . . . 8  |-  ( ( -.  ph  \/  ch ) 
<->  ( ch  \/  -.  ph ) )
1311, 12bitri 252 . . . . . . 7  |-  ( (
ph  ->  ch )  <->  ( ch  \/  -.  ph ) )
1413imbi2i 313 . . . . . 6  |-  ( ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )  <->  ( ( ph  ->  ps )  -> 
( ch  \/  -.  ph ) ) )
1510, 14bitri 252 . . . . 5  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ( ph  ->  ps )  -> 
( ch  \/  -.  ph ) ) )
169, 15bitri 252 . . . 4  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ( ph  ->  ps )  -> 
( ch  \/  -.  ph ) ) )
17 pm2.21 111 . . . . . 6  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
1817olcd 394 . . . . 5  |-  ( -. 
ph  ->  ( ch  \/  ( ph  ->  ps )
) )
1918biantrur 508 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( ch  \/  -.  ph ) )  <->  ( ( -.  ph  ->  ( ch  \/  ( ph  ->  ps ) ) )  /\  ( ( ph  ->  ps )  ->  ( ch  \/  -.  ph ) ) ) )
2016, 19bitri 252 . . 3  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ( -.  ph  ->  ( ch  \/  ( ph  ->  ps ) ) )  /\  ( ( ph  ->  ps )  ->  ( ch  \/  -.  ph ) ) ) )
218, 20anbi12i 701 . 2  |-  ( ( ( ch  ->  ( ph  \/  ps ) )  /\  ( ( ph  /\ 
ps )  ->  ch ) )  <->  ( (
( ( ch  /\  ps )  ->  ( ph  ->  ps ) )  /\  ( ( ch  /\  ( ph  ->  ps )
)  ->  ps )
)  /\  ( ( -.  ph  ->  ( ch  \/  ( ph  ->  ps ) ) )  /\  ( ( ph  ->  ps )  ->  ( ch  \/  -.  ph ) ) ) ) )
22 ancom 451 . 2  |-  ( ( ( ch  ->  ( ph  \/  ps ) )  /\  ( ( ph  /\ 
ps )  ->  ch ) )  <->  ( (
( ph  /\  ps )  ->  ch )  /\  ( ch  ->  ( ph  \/  ps ) ) ) )
231, 21, 223bitr2i 276 1  |-  ( ( ( ph  ->  ps ) 
<-> if- ( ch ,  ps ,  -.  ph ) )  <-> 
( ( ( ph  /\ 
ps )  ->  ch )  /\  ( ch  ->  (
ph  \/  ps )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370  if-wif 1420
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  df-ifp 1421
This theorem is referenced by:  ifpim3  36110  ifpim4  36112
  Copyright terms: Public domain W3C validator