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Theorem ifpimimb 36074
Description: Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
Assertion
Ref Expression
ifpimimb  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta ) ) )

Proof of Theorem ifpimimb
StepHypRef Expression
1 dfifp2 1423 . 2  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <->  ( ( ph  ->  ( ps  ->  ch ) )  /\  ( -.  ph  ->  ( th  ->  ta ) ) ) )
2 imor 414 . . . 4  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( -.  ph  \/  ( ps  ->  ch ) ) )
3 pm4.8 367 . . . . . 6  |-  ( (
ph  ->  -.  ph )  <->  -.  ph )
43bicomi 206 . . . . 5  |-  ( -. 
ph 
<->  ( ph  ->  -.  ph ) )
54orbi1i 523 . . . 4  |-  ( ( -.  ph  \/  ( ps  ->  ch ) )  <-> 
( ( ph  ->  -. 
ph )  \/  ( ps  ->  ch ) ) )
6 id 23 . . . . . 6  |-  ( ph  ->  ph )
76orci 392 . . . . 5  |-  ( (
ph  ->  ph )  \/  ( th  ->  ch ) )
87biantru 508 . . . 4  |-  ( ( ( ph  ->  -.  ph )  \/  ( ps 
->  ch ) )  <->  ( (
( ph  ->  -.  ph )  \/  ( ps  ->  ch ) )  /\  ( ( ph  ->  ph )  \/  ( th 
->  ch ) ) ) )
92, 5, 83bitri 275 . . 3  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( (
( ph  ->  -.  ph )  \/  ( ps  ->  ch ) )  /\  ( ( ph  ->  ph )  \/  ( th 
->  ch ) ) ) )
10 pm4.64 374 . . . 4  |-  ( ( -.  ph  ->  ( th 
->  ta ) )  <->  ( ph  \/  ( th  ->  ta ) ) )
11 pm4.81 368 . . . . . 6  |-  ( ( -.  ph  ->  ph )  <->  ph )
1211bicomi 206 . . . . 5  |-  ( ph  <->  ( -.  ph  ->  ph )
)
1312orbi1i 523 . . . 4  |-  ( (
ph  \/  ( th  ->  ta ) )  <->  ( ( -.  ph  ->  ph )  \/  ( th  ->  ta ) ) )
146orci 392 . . . . 5  |-  ( (
ph  ->  ph )  \/  ( ps  ->  ta ) )
1514biantrur 509 . . . 4  |-  ( ( ( -.  ph  ->  ph )  \/  ( th 
->  ta ) )  <->  ( (
( ph  ->  ph )  \/  ( ps  ->  ta ) )  /\  (
( -.  ph  ->  ph )  \/  ( th 
->  ta ) ) ) )
1610, 13, 153bitri 275 . . 3  |-  ( ( -.  ph  ->  ( th 
->  ta ) )  <->  ( (
( ph  ->  ph )  \/  ( ps  ->  ta ) )  /\  (
( -.  ph  ->  ph )  \/  ( th 
->  ta ) ) ) )
179, 16anbi12i 702 . 2  |-  ( ( ( ph  ->  ( ps  ->  ch ) )  /\  ( -.  ph  ->  ( th  ->  ta ) ) )  <->  ( (
( ( ph  ->  -. 
ph )  \/  ( ps  ->  ch ) )  /\  ( ( ph  ->  ph )  \/  ( th  ->  ch ) ) )  /\  ( ( ( ph  ->  ph )  \/  ( ps  ->  ta ) )  /\  (
( -.  ph  ->  ph )  \/  ( th 
->  ta ) ) ) ) )
18 ifpim123g 36070 . . 3  |-  ( (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
)  <->  ( ( ( ( ph  ->  -.  ph )  \/  ( ps 
->  ch ) )  /\  ( ( ph  ->  ph )  \/  ( th 
->  ch ) ) )  /\  ( ( (
ph  ->  ph )  \/  ( ps  ->  ta ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) ) ) )
1918bicomi 206 . 2  |-  ( ( ( ( ( ph  ->  -.  ph )  \/  ( ps  ->  ch ) )  /\  (
( ph  ->  ph )  \/  ( th  ->  ch ) ) )  /\  ( ( ( ph  ->  ph )  \/  ( ps  ->  ta ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) ) )  <->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
) )
201, 17, 193bitri 275 1  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371  if-wif 1421
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 189  df-or 372  df-an 373  df-ifp 1422
This theorem is referenced by:  ifpororb  36075  ifpbibib  36080
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