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Theorem ifpan23 35802
Description: Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpan23  |-  ( (if- ( ph ,  ps ,  ch )  /\ if- ( ph ,  th ,  ta ) )  <-> if- ( ph , 
( ps  /\  th ) ,  ( ch  /\ 
ta ) ) )

Proof of Theorem ifpan23
StepHypRef Expression
1 ifpan123g 35801 . 2  |-  ( (if- ( ph ,  ps ,  ch )  /\ if- ( ph ,  th ,  ta ) )  <->  ( (
( -.  ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ( -. 
ph  \/  th )  /\  ( ph  \/  ta ) ) ) )
2 an4 831 . 2  |-  ( ( ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ( -. 
ph  \/  th )  /\  ( ph  \/  ta ) ) )  <->  ( (
( -.  ph  \/  ps )  /\  ( -.  ph  \/  th )
)  /\  ( ( ph  \/  ch )  /\  ( ph  \/  ta )
) ) )
3 dfifp4 1424 . . 3  |-  (if- (
ph ,  ( ps 
/\  th ) ,  ( ch  /\  ta )
)  <->  ( ( -. 
ph  \/  ( ps  /\ 
th ) )  /\  ( ph  \/  ( ch 
/\  ta ) ) ) )
4 ordi 872 . . . 4  |-  ( ( -.  ph  \/  ( ps  /\  th ) )  <-> 
( ( -.  ph  \/  ps )  /\  ( -.  ph  \/  th )
) )
5 ordi 872 . . . 4  |-  ( (
ph  \/  ( ch  /\ 
ta ) )  <->  ( ( ph  \/  ch )  /\  ( ph  \/  ta )
) )
64, 5anbi12i 701 . . 3  |-  ( ( ( -.  ph  \/  ( ps  /\  th )
)  /\  ( ph  \/  ( ch  /\  ta ) ) )  <->  ( (
( -.  ph  \/  ps )  /\  ( -.  ph  \/  th )
)  /\  ( ( ph  \/  ch )  /\  ( ph  \/  ta )
) ) )
73, 6bitr2i 253 . 2  |-  ( ( ( ( -.  ph  \/  ps )  /\  ( -.  ph  \/  th )
)  /\  ( ( ph  \/  ch )  /\  ( ph  \/  ta )
) )  <-> if- ( ph , 
( ps  /\  th ) ,  ( ch  /\ 
ta ) ) )
81, 2, 73bitri 274 1  |-  ( (if- ( ph ,  ps ,  ch )  /\ if- ( ph ,  th ,  ta ) )  <-> if- ( ph , 
( ps  /\  th ) ,  ( ch  /\ 
ta ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> 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:  ifpdfbi  35816  ifpdfxor  35830
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