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Theorem ifpororb 36119
Description: Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
Assertion
Ref Expression
ifpororb  |-  (if- (
ph ,  ( ps  \/  ch ) ,  ( th  \/  ta ) )  <->  (if- ( ph ,  ps ,  th )  \/ if- ( ph ,  ch ,  ta )
) )

Proof of Theorem ifpororb
StepHypRef Expression
1 dfifp2 1422 . 2  |-  (if- (
ph ,  ( ps  \/  ch ) ,  ( th  \/  ta ) )  <->  ( ( ph  ->  ( ps  \/  ch ) )  /\  ( -.  ph  ->  ( th  \/  ta ) ) ) )
2 df-or 371 . . . 4  |-  ( ( ps  \/  ch )  <->  ( -.  ps  ->  ch ) )
32imbi2i 313 . . 3  |-  ( (
ph  ->  ( ps  \/  ch ) )  <->  ( ph  ->  ( -.  ps  ->  ch ) ) )
4 df-or 371 . . . 4  |-  ( ( th  \/  ta )  <->  ( -.  th  ->  ta ) )
54imbi2i 313 . . 3  |-  ( ( -.  ph  ->  ( th  \/  ta ) )  <-> 
( -.  ph  ->  ( -.  th  ->  ta ) ) )
63, 5anbi12i 701 . 2  |-  ( ( ( ph  ->  ( ps  \/  ch ) )  /\  ( -.  ph  ->  ( th  \/  ta ) ) )  <->  ( ( ph  ->  ( -.  ps  ->  ch ) )  /\  ( -.  ph  ->  ( -.  th  ->  ta )
) ) )
7 ifpimimb 36118 . . 3  |-  (if- (
ph ,  ( -. 
ps  ->  ch ) ,  ( -.  th  ->  ta ) )  <->  (if- ( ph ,  -.  ps ,  -.  th )  -> if- ( ph ,  ch ,  ta )
) )
8 dfifp2 1422 . . 3  |-  (if- (
ph ,  ( -. 
ps  ->  ch ) ,  ( -.  th  ->  ta ) )  <->  ( ( ph  ->  ( -.  ps  ->  ch ) )  /\  ( -.  ph  ->  ( -.  th  ->  ta )
) ) )
9 imor 413 . . . 4  |-  ( (if- ( ph ,  -.  ps ,  -.  th )  -> if- ( ph ,  ch ,  ta ) )  <->  ( -. if- ( ph ,  -.  ps ,  -.  th )  \/ if- ( ph ,  ch ,  ta ) ) )
10 ifpnot23d 36099 . . . . 5  |-  ( -. if- ( ph ,  -.  ps ,  -.  th )  <-> if- (
ph ,  ps ,  th ) )
1110orbi1i 522 . . . 4  |-  ( ( -. if- ( ph ,  -.  ps ,  -.  th )  \/ if- ( ph ,  ch ,  ta )
)  <->  (if- ( ph ,  ps ,  th )  \/ if- ( ph ,  ch ,  ta ) ) )
129, 11bitri 252 . . 3  |-  ( (if- ( ph ,  -.  ps ,  -.  th )  -> if- ( ph ,  ch ,  ta ) )  <->  (if- ( ph ,  ps ,  th )  \/ if- ( ph ,  ch ,  ta )
) )
137, 8, 123bitr3i 278 . 2  |-  ( ( ( ph  ->  ( -.  ps  ->  ch )
)  /\  ( -.  ph 
->  ( -.  th  ->  ta ) ) )  <->  (if- ( ph ,  ps ,  th )  \/ if- ( ph ,  ch ,  ta )
) )
141, 6, 133bitri 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    \/ 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:  ifpananb  36120
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