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Theorem ifpbibib 36124
Description: Factor conditional logic operator over biimplication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
Assertion
Ref Expression
ifpbibib  |-  (if- (
ph ,  ( ps  <->  ch ) ,  ( th  <->  ta ) )  <->  (if- ( ph ,  ps ,  th ) 
<-> if- ( ph ,  ch ,  ta ) ) )

Proof of Theorem ifpbibib
StepHypRef Expression
1 dfifp2 1422 . 2  |-  (if- (
ph ,  ( ps  <->  ch ) ,  ( th  <->  ta ) )  <->  ( ( ph  ->  ( ps  <->  ch )
)  /\  ( -.  ph 
->  ( th  <->  ta )
) ) )
2 dfbi2 632 . . . . . 6  |-  ( ( ps  <->  ch )  <->  ( ( ps  ->  ch )  /\  ( ch  ->  ps )
) )
32imbi2i 313 . . . . 5  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ph  ->  ( ( ps  ->  ch )  /\  ( ch  ->  ps ) ) ) )
4 jcab 871 . . . . 5  |-  ( (
ph  ->  ( ( ps 
->  ch )  /\  ( ch  ->  ps ) ) )  <->  ( ( ph  ->  ( ps  ->  ch ) )  /\  ( ph  ->  ( ch  ->  ps ) ) ) )
53, 4bitri 252 . . . 4  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  ->  ( ps  ->  ch ) )  /\  ( ph  ->  ( ch  ->  ps ) ) ) )
6 dfbi2 632 . . . . . 6  |-  ( ( th  <->  ta )  <->  ( ( th  ->  ta )  /\  ( ta  ->  th )
) )
76imbi2i 313 . . . . 5  |-  ( ( -.  ph  ->  ( th  <->  ta ) )  <->  ( -.  ph 
->  ( ( th  ->  ta )  /\  ( ta 
->  th ) ) ) )
8 jcab 871 . . . . 5  |-  ( ( -.  ph  ->  ( ( th  ->  ta )  /\  ( ta  ->  th )
) )  <->  ( ( -.  ph  ->  ( th  ->  ta ) )  /\  ( -.  ph  ->  ( ta  ->  th ) ) ) )
97, 8bitri 252 . . . 4  |-  ( ( -.  ph  ->  ( th  <->  ta ) )  <->  ( ( -.  ph  ->  ( th  ->  ta ) )  /\  ( -.  ph  ->  ( ta  ->  th ) ) ) )
105, 9anbi12i 701 . . 3  |-  ( ( ( ph  ->  ( ps 
<->  ch ) )  /\  ( -.  ph  ->  ( th 
<->  ta ) ) )  <-> 
( ( ( ph  ->  ( ps  ->  ch ) )  /\  ( ph  ->  ( ch  ->  ps ) ) )  /\  ( ( -.  ph  ->  ( th  ->  ta ) )  /\  ( -.  ph  ->  ( ta  ->  th ) ) ) ) )
11 an4 831 . . 3  |-  ( ( ( ( ph  ->  ( ps  ->  ch )
)  /\  ( ph  ->  ( ch  ->  ps ) ) )  /\  ( ( -.  ph  ->  ( th  ->  ta ) )  /\  ( -.  ph  ->  ( ta  ->  th ) ) ) )  <->  ( ( (
ph  ->  ( ps  ->  ch ) )  /\  ( -.  ph  ->  ( th  ->  ta ) ) )  /\  ( ( ph  ->  ( ch  ->  ps ) )  /\  ( -.  ph  ->  ( ta  ->  th ) ) ) ) )
1210, 11bitri 252 . 2  |-  ( ( ( ph  ->  ( ps 
<->  ch ) )  /\  ( -.  ph  ->  ( th 
<->  ta ) ) )  <-> 
( ( ( ph  ->  ( ps  ->  ch ) )  /\  ( -.  ph  ->  ( th  ->  ta ) ) )  /\  ( ( ph  ->  ( ch  ->  ps ) )  /\  ( -.  ph  ->  ( ta  ->  th ) ) ) ) )
13 dfifp2 1422 . . . . 5  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <->  ( ( ph  ->  ( ps  ->  ch ) )  /\  ( -.  ph  ->  ( th  ->  ta ) ) ) )
14 ifpimimb 36118 . . . . 5  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta ) ) )
1513, 14bitr3i 254 . . . 4  |-  ( ( ( ph  ->  ( ps  ->  ch ) )  /\  ( -.  ph  ->  ( th  ->  ta ) ) )  <->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
) )
16 dfifp2 1422 . . . . 5  |-  (if- (
ph ,  ( ch 
->  ps ) ,  ( ta  ->  th )
)  <->  ( ( ph  ->  ( ch  ->  ps ) )  /\  ( -.  ph  ->  ( ta  ->  th ) ) ) )
17 ifpimimb 36118 . . . . 5  |-  (if- (
ph ,  ( ch 
->  ps ) ,  ( ta  ->  th )
)  <->  (if- ( ph ,  ch ,  ta )  -> if- ( ph ,  ps ,  th ) ) )
1816, 17bitr3i 254 . . . 4  |-  ( ( ( ph  ->  ( ch  ->  ps ) )  /\  ( -.  ph  ->  ( ta  ->  th )
) )  <->  (if- ( ph ,  ch ,  ta )  -> if- ( ph ,  ps ,  th )
) )
1915, 18anbi12i 701 . . 3  |-  ( ( ( ( ph  ->  ( ps  ->  ch )
)  /\  ( -.  ph 
->  ( th  ->  ta ) ) )  /\  ( ( ph  ->  ( ch  ->  ps )
)  /\  ( -.  ph 
->  ( ta  ->  th )
) ) )  <->  ( (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
)  /\  (if- ( ph ,  ch ,  ta )  -> if- ( ph ,  ps ,  th )
) ) )
20 dfbi2 632 . . 3  |-  ( (if- ( ph ,  ps ,  th )  <-> if- ( ph ,  ch ,  ta )
)  <->  ( (if- (
ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
)  /\  (if- ( ph ,  ch ,  ta )  -> if- ( ph ,  ps ,  th )
) ) )
2119, 20bitr4i 255 . 2  |-  ( ( ( ( ph  ->  ( ps  ->  ch )
)  /\  ( -.  ph 
->  ( th  ->  ta ) ) )  /\  ( ( ph  ->  ( ch  ->  ps )
)  /\  ( -.  ph 
->  ( ta  ->  th )
) ) )  <->  (if- ( ph ,  ps ,  th ) 
<-> if- ( ph ,  ch ,  ta ) ) )
221, 12, 213bitri 274 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 187    /\ 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:  ifpxorxorb  36125
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