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Theorem ifpid1g 38122
Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpid1g  |-  ( (
ph 
<-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ch  ->  ph )  /\  ( ph  ->  ps ) ) )

Proof of Theorem ifpid1g
StepHypRef Expression
1 ifpidg 38121 . 2  |-  ( (
ph 
<-> if- ( ph ,  ps ,  ch ) )  <->  ( (
( ( ph  /\  ps )  ->  ph )  /\  ( ( ph  /\  ph )  ->  ps )
)  /\  ( ( ch  ->  ( ph  \/  ph ) )  /\  ( ph  ->  ( ph  \/  ch ) ) ) ) )
2 ancom 448 . 2  |-  ( ( ( ( ( ph  /\ 
ps )  ->  ph )  /\  ( ( ph  /\  ph )  ->  ps )
)  /\  ( ( ch  ->  ( ph  \/  ph ) )  /\  ( ph  ->  ( ph  \/  ch ) ) ) )  <-> 
( ( ( ch 
->  ( ph  \/  ph ) )  /\  ( ph  ->  ( ph  \/  ch ) ) )  /\  ( ( ( ph  /\ 
ps )  ->  ph )  /\  ( ( ph  /\  ph )  ->  ps )
) ) )
3 pm4.25 513 . . . . 5  |-  ( ph  <->  (
ph  \/  ph ) )
43imbi2i 310 . . . 4  |-  ( ( ch  ->  ph )  <->  ( ch  ->  ( ph  \/  ph ) ) )
5 orc 383 . . . . 5  |-  ( ph  ->  ( ph  \/  ch ) )
65biantru 503 . . . 4  |-  ( ( ch  ->  ( ph  \/  ph ) )  <->  ( ( ch  ->  ( ph  \/  ph ) )  /\  ( ph  ->  ( ph  \/  ch ) ) ) )
74, 6bitr2i 250 . . 3  |-  ( ( ( ch  ->  ( ph  \/  ph ) )  /\  ( ph  ->  (
ph  \/  ch )
) )  <->  ( ch  ->  ph ) )
8 pm4.24 641 . . . . 5  |-  ( ph  <->  (
ph  /\  ph ) )
98imbi1i 323 . . . 4  |-  ( (
ph  ->  ps )  <->  ( ( ph  /\  ph )  ->  ps ) )
10 simpl 455 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
1110biantrur 504 . . . 4  |-  ( ( ( ph  /\  ph )  ->  ps )  <->  ( (
( ph  /\  ps )  ->  ph )  /\  (
( ph  /\  ph )  ->  ps ) ) )
129, 11bitr2i 250 . . 3  |-  ( ( ( ( ph  /\  ps )  ->  ph )  /\  ( ( ph  /\  ph )  ->  ps )
)  <->  ( ph  ->  ps ) )
137, 12anbi12i 695 . 2  |-  ( ( ( ( ch  ->  (
ph  \/  ph ) )  /\  ( ph  ->  (
ph  \/  ch )
) )  /\  (
( ( ph  /\  ps )  ->  ph )  /\  ( ( ph  /\  ph )  ->  ps )
) )  <->  ( ( ch  ->  ph )  /\  ( ph  ->  ps ) ) )
141, 2, 133bitri 271 1  |-  ( (
ph 
<-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ch  ->  ph )  /\  ( ph  ->  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367  if-wif 1382
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 368  df-an 369  df-ifp 1383
This theorem is referenced by: (None)
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