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Theorem ifpidg 36105
Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpidg  |-  ( ( th  <-> if- ( ph ,  ps ,  ch ) )  <->  ( (
( ( ph  /\  ps )  ->  th )  /\  ( ( ph  /\  th )  ->  ps )
)  /\  ( ( ch  ->  ( ph  \/  th ) )  /\  ( th  ->  ( ph  \/  ch ) ) ) ) )

Proof of Theorem ifpidg
StepHypRef Expression
1 dfifp4 1424 . . 3  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( -. 
ph  \/  ps )  /\  ( ph  \/  ch ) ) )
21bibi2i 314 . 2  |-  ( ( th  <-> if- ( ph ,  ps ,  ch ) )  <->  ( th  <->  ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) ) )
3 dfbi2 632 . . 3  |-  ( ( th  <->  ( ( -. 
ph  \/  ps )  /\  ( ph  \/  ch ) ) )  <->  ( ( th  ->  ( ( -. 
ph  \/  ps )  /\  ( ph  \/  ch ) ) )  /\  ( ( ( -. 
ph  \/  ps )  /\  ( ph  \/  ch ) )  ->  th )
) )
4 imor 413 . . . . 5  |-  ( ( th  ->  ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) )  <->  ( -.  th  \/  ( ( -. 
ph  \/  ps )  /\  ( ph  \/  ch ) ) ) )
5 ordi 872 . . . . 5  |-  ( ( -.  th  \/  (
( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) )  <->  ( ( -. 
th  \/  ( -.  ph  \/  ps ) )  /\  ( -.  th  \/  ( ph  \/  ch ) ) ) )
6 ancomst 453 . . . . . . 7  |-  ( ( ( ph  /\  th )  ->  ps )  <->  ( ( th  /\  ph )  ->  ps ) )
7 impexp 447 . . . . . . 7  |-  ( ( ( th  /\  ph )  ->  ps )  <->  ( th  ->  ( ph  ->  ps ) ) )
8 imor 413 . . . . . . . . 9  |-  ( (
ph  ->  ps )  <->  ( -.  ph  \/  ps ) )
98imbi2i 313 . . . . . . . 8  |-  ( ( th  ->  ( ph  ->  ps ) )  <->  ( th  ->  ( -.  ph  \/  ps ) ) )
10 imor 413 . . . . . . . 8  |-  ( ( th  ->  ( -.  ph  \/  ps ) )  <-> 
( -.  th  \/  ( -.  ph  \/  ps ) ) )
119, 10bitri 252 . . . . . . 7  |-  ( ( th  ->  ( ph  ->  ps ) )  <->  ( -.  th  \/  ( -.  ph  \/  ps ) ) )
126, 7, 113bitrri 275 . . . . . 6  |-  ( ( -.  th  \/  ( -.  ph  \/  ps )
)  <->  ( ( ph  /\ 
th )  ->  ps ) )
13 imor 413 . . . . . . 7  |-  ( ( th  ->  ( ph  \/  ch ) )  <->  ( -.  th  \/  ( ph  \/  ch ) ) )
1413bicomi 205 . . . . . 6  |-  ( ( -.  th  \/  ( ph  \/  ch ) )  <-> 
( th  ->  ( ph  \/  ch ) ) )
1512, 14anbi12i 701 . . . . 5  |-  ( ( ( -.  th  \/  ( -.  ph  \/  ps ) )  /\  ( -.  th  \/  ( ph  \/  ch ) ) )  <-> 
( ( ( ph  /\ 
th )  ->  ps )  /\  ( th  ->  (
ph  \/  ch )
) ) )
164, 5, 153bitri 274 . . . 4  |-  ( ( th  ->  ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) )  <->  ( (
( ph  /\  th )  ->  ps )  /\  ( th  ->  ( ph  \/  ch ) ) ) )
178bicomi 205 . . . . . . . 8  |-  ( ( -.  ph  \/  ps ) 
<->  ( ph  ->  ps ) )
18 df-or 371 . . . . . . . 8  |-  ( (
ph  \/  ch )  <->  ( -.  ph  ->  ch )
)
1917, 18anbi12i 701 . . . . . . 7  |-  ( ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) )  <-> 
( ( ph  ->  ps )  /\  ( -. 
ph  ->  ch ) ) )
20 cases2 980 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )
2120bicomi 205 . . . . . . 7  |-  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch ) ) )
2219, 21bitri 252 . . . . . 6  |-  ( ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) )  <-> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
2322imbi1i 326 . . . . 5  |-  ( ( ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) )  ->  th )  <->  ( (
( ph  /\  ps )  \/  ( -.  ph  /\  ch ) )  ->  th )
)
24 jaob 790 . . . . 5  |-  ( ( ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
)  ->  th )  <->  ( ( ( ph  /\  ps )  ->  th )  /\  ( ( -.  ph  /\ 
ch )  ->  th )
) )
25 ancomst 453 . . . . . . 7  |-  ( ( ( -.  ph  /\  ch )  ->  th )  <->  ( ( ch  /\  -.  ph )  ->  th )
)
26 pm5.6 920 . . . . . . 7  |-  ( ( ( ch  /\  -.  ph )  ->  th )  <->  ( ch  ->  ( ph  \/  th ) ) )
2725, 26bitri 252 . . . . . 6  |-  ( ( ( -.  ph  /\  ch )  ->  th )  <->  ( ch  ->  ( ph  \/  th ) ) )
2827anbi2i 698 . . . . 5  |-  ( ( ( ( ph  /\  ps )  ->  th )  /\  ( ( -.  ph  /\ 
ch )  ->  th )
)  <->  ( ( (
ph  /\  ps )  ->  th )  /\  ( ch  ->  ( ph  \/  th ) ) ) )
2923, 24, 283bitri 274 . . . 4  |-  ( ( ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) )  ->  th )  <->  ( (
( ph  /\  ps )  ->  th )  /\  ( ch  ->  ( ph  \/  th ) ) ) )
3016, 29anbi12i 701 . . 3  |-  ( ( ( th  ->  (
( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) )  /\  ( ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) )  ->  th ) )  <->  ( (
( ( ph  /\  th )  ->  ps )  /\  ( th  ->  ( ph  \/  ch ) ) )  /\  ( ( ( ph  /\  ps )  ->  th )  /\  ( ch  ->  ( ph  \/  th ) ) ) ) )
313, 30bitri 252 . 2  |-  ( ( th  <->  ( ( -. 
ph  \/  ps )  /\  ( ph  \/  ch ) ) )  <->  ( (
( ( ph  /\  th )  ->  ps )  /\  ( th  ->  ( ph  \/  ch ) ) )  /\  ( ( ( ph  /\  ps )  ->  th )  /\  ( ch  ->  ( ph  \/  th ) ) ) ) )
32 ancom 451 . . 3  |-  ( ( ( ( ( ph  /\ 
th )  ->  ps )  /\  ( th  ->  (
ph  \/  ch )
) )  /\  (
( ( ph  /\  ps )  ->  th )  /\  ( ch  ->  ( ph  \/  th ) ) ) )  <->  ( (
( ( ph  /\  ps )  ->  th )  /\  ( ch  ->  ( ph  \/  th ) ) )  /\  ( ( ( ph  /\  th )  ->  ps )  /\  ( th  ->  ( ph  \/  ch ) ) ) ) )
33 an4 831 . . 3  |-  ( ( ( ( ( ph  /\ 
ps )  ->  th )  /\  ( ch  ->  ( ph  \/  th ) ) )  /\  ( ( ( ph  /\  th )  ->  ps )  /\  ( th  ->  ( ph  \/  ch ) ) ) )  <->  ( ( ( ( ph  /\  ps )  ->  th )  /\  (
( ph  /\  th )  ->  ps ) )  /\  ( ( ch  ->  (
ph  \/  th )
)  /\  ( th  ->  ( ph  \/  ch ) ) ) ) )
3432, 33bitri 252 . 2  |-  ( ( ( ( ( ph  /\ 
th )  ->  ps )  /\  ( th  ->  (
ph  \/  ch )
) )  /\  (
( ( ph  /\  ps )  ->  th )  /\  ( ch  ->  ( ph  \/  th ) ) ) )  <->  ( (
( ( ph  /\  ps )  ->  th )  /\  ( ( ph  /\  th )  ->  ps )
)  /\  ( ( ch  ->  ( ph  \/  th ) )  /\  ( th  ->  ( ph  \/  ch ) ) ) ) )
352, 31, 343bitri 274 1  |-  ( ( th  <-> if- ( ph ,  ps ,  ch ) )  <->  ( (
( ( ph  /\  ps )  ->  th )  /\  ( ( ph  /\  th )  ->  ps )
)  /\  ( ( ch  ->  ( ph  \/  th ) )  /\  ( th  ->  ( ph  \/  ch ) ) ) ) )
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:  ifpid3g  36106  ifpid2g  36107  ifpid1g  36108  ifpim23g  36109
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