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Theorem ifpananb 36221
Description: Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
Assertion
Ref Expression
ifpananb  |-  (if- (
ph ,  ( ps 
/\  ch ) ,  ( th  /\  ta )
)  <->  (if- ( ph ,  ps ,  th )  /\ if- ( ph ,  ch ,  ta ) ) )

Proof of Theorem ifpananb
StepHypRef Expression
1 anor 497 . . 3  |-  ( ( ps  /\  ch )  <->  -.  ( -.  ps  \/  -.  ch ) )
2 anor 497 . . 3  |-  ( ( th  /\  ta )  <->  -.  ( -.  th  \/  -.  ta ) )
3 ifpbi23 36187 . . 3  |-  ( ( ( ( ps  /\  ch )  <->  -.  ( -.  ps  \/  -.  ch )
)  /\  ( ( th  /\  ta )  <->  -.  ( -.  th  \/  -.  ta ) ) )  -> 
(if- ( ph , 
( ps  /\  ch ) ,  ( th  /\  ta ) )  <-> if- ( ph ,  -.  ( -.  ps  \/  -.  ch ) ,  -.  ( -.  th  \/  -.  ta ) ) ) )
41, 2, 3mp2an 686 . 2  |-  (if- (
ph ,  ( ps 
/\  ch ) ,  ( th  /\  ta )
)  <-> if- ( ph ,  -.  ( -.  ps  \/  -.  ch ) ,  -.  ( -.  th  \/  -.  ta ) ) )
5 ifpororb 36220 . . . . 5  |-  (if- (
ph ,  ( -. 
ps  \/  -.  ch ) ,  ( -.  th  \/  -.  ta ) )  <-> 
(if- ( ph ,  -.  ps ,  -.  th )  \/ if- ( ph ,  -.  ch ,  -.  ta ) ) )
6 ifpnotnotb 36194 . . . . . 6  |-  (if- (
ph ,  -.  ps ,  -.  th )  <->  -. if- ( ph ,  ps ,  th )
)
7 ifpnotnotb 36194 . . . . . 6  |-  (if- (
ph ,  -.  ch ,  -.  ta )  <->  -. if- ( ph ,  ch ,  ta )
)
86, 7orbi12i 530 . . . . 5  |-  ( (if- ( ph ,  -.  ps ,  -.  th )  \/ if- ( ph ,  -.  ch ,  -.  ta )
)  <->  ( -. if- ( ph ,  ps ,  th )  \/  -. if- ( ph ,  ch ,  ta )
) )
95, 8bitri 257 . . . 4  |-  (if- (
ph ,  ( -. 
ps  \/  -.  ch ) ,  ( -.  th  \/  -.  ta ) )  <-> 
( -. if- ( ph ,  ps ,  th )  \/  -. if- ( ph ,  ch ,  ta )
) )
109notbii 303 . . 3  |-  ( -. if- ( ph ,  ( -.  ps  \/  -.  ch ) ,  ( -. 
th  \/  -.  ta )
)  <->  -.  ( -. if- ( ph ,  ps ,  th )  \/  -. if- ( ph ,  ch ,  ta ) ) )
11 ifpnotnotb 36194 . . 3  |-  (if- (
ph ,  -.  ( -.  ps  \/  -.  ch ) ,  -.  ( -.  th  \/  -.  ta ) )  <->  -. if- ( ph ,  ( -.  ps  \/  -.  ch ) ,  ( -.  th  \/  -.  ta ) ) )
12 anor 497 . . 3  |-  ( (if- ( ph ,  ps ,  th )  /\ if- ( ph ,  ch ,  ta ) )  <->  -.  ( -. if- ( ph ,  ps ,  th )  \/  -. if- ( ph ,  ch ,  ta ) ) )
1310, 11, 123bitr4i 285 . 2  |-  (if- (
ph ,  -.  ( -.  ps  \/  -.  ch ) ,  -.  ( -.  th  \/  -.  ta ) )  <->  (if- ( ph ,  ps ,  th )  /\ if- ( ph ,  ch ,  ta )
) )
144, 13bitri 257 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    <-> wb 189    \/ wo 375    /\ wa 376  if-wif 983
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 190  df-or 377  df-an 378  df-ifp 984
This theorem is referenced by:  ifpnannanb  36222
  Copyright terms: Public domain W3C validator