Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpor123g Structured version   Unicode version

Theorem ifpor123g 36072
Description: Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
Assertion
Ref Expression
ifpor123g  |-  ( (if- ( ph ,  ch ,  ta )  \/ if- ( ps ,  th ,  et ) )  <->  ( (
( ( ph  ->  -. 
ps )  \/  ( ch  \/  th ) )  /\  ( ( ps 
->  ph )  \/  ( ta  \/  th ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  \/  et ) )  /\  ( ( -.  ps  ->  ph )  \/  ( ta  \/  et ) ) ) ) )

Proof of Theorem ifpor123g
StepHypRef Expression
1 df-or 371 . . . 4  |-  ( (if- ( ph ,  ch ,  ta )  \/ if- ( ps ,  th ,  et ) )  <->  ( -. if- ( ph ,  ch ,  ta )  -> if- ( ps ,  th ,  et ) ) )
2 ifpnot23 36042 . . . . 5  |-  ( -. if- ( ph ,  ch ,  ta )  <-> if- ( ph ,  -.  ch ,  -.  ta ) )
32imbi1i 326 . . . 4  |-  ( ( -. if- ( ph ,  ch ,  ta )  -> if- ( ps ,  th ,  et ) )  <->  (if- ( ph ,  -.  ch ,  -.  ta )  -> if- ( ps ,  th ,  et ) ) )
41, 3bitri 252 . . 3  |-  ( (if- ( ph ,  ch ,  ta )  \/ if- ( ps ,  th ,  et ) )  <->  (if- ( ph ,  -.  ch ,  -.  ta )  -> if- ( ps ,  th ,  et ) ) )
5 ifpim123g 36064 . . 3  |-  ( (if- ( ph ,  -.  ch ,  -.  ta )  -> if- ( ps ,  th ,  et ) )  <->  ( (
( ( ph  ->  -. 
ps )  \/  ( -.  ch  ->  th )
)  /\  ( ( ps  ->  ph )  \/  ( -.  ta  ->  th )
) )  /\  (
( ( ph  ->  ps )  \/  ( -. 
ch  ->  et ) )  /\  ( ( -. 
ps  ->  ph )  \/  ( -.  ta  ->  et )
) ) ) )
64, 5bitri 252 . 2  |-  ( (if- ( ph ,  ch ,  ta )  \/ if- ( ps ,  th ,  et ) )  <->  ( (
( ( ph  ->  -. 
ps )  \/  ( -.  ch  ->  th )
)  /\  ( ( ps  ->  ph )  \/  ( -.  ta  ->  th )
) )  /\  (
( ( ph  ->  ps )  \/  ( -. 
ch  ->  et ) )  /\  ( ( -. 
ps  ->  ph )  \/  ( -.  ta  ->  et )
) ) ) )
7 pm4.64 373 . . . . 5  |-  ( ( -.  ch  ->  th )  <->  ( ch  \/  th )
)
87orbi2i 521 . . . 4  |-  ( ( ( ph  ->  -.  ps )  \/  ( -.  ch  ->  th )
)  <->  ( ( ph  ->  -.  ps )  \/  ( ch  \/  th ) ) )
9 pm4.64 373 . . . . 5  |-  ( ( -.  ta  ->  th )  <->  ( ta  \/  th )
)
109orbi2i 521 . . . 4  |-  ( ( ( ps  ->  ph )  \/  ( -.  ta  ->  th ) )  <->  ( ( ps  ->  ph )  \/  ( ta  \/  th ) ) )
118, 10anbi12i 701 . . 3  |-  ( ( ( ( ph  ->  -. 
ps )  \/  ( -.  ch  ->  th )
)  /\  ( ( ps  ->  ph )  \/  ( -.  ta  ->  th )
) )  <->  ( (
( ph  ->  -.  ps )  \/  ( ch  \/  th ) )  /\  ( ( ps  ->  ph )  \/  ( ta  \/  th ) ) ) )
12 pm4.64 373 . . . . 5  |-  ( ( -.  ch  ->  et ) 
<->  ( ch  \/  et ) )
1312orbi2i 521 . . . 4  |-  ( ( ( ph  ->  ps )  \/  ( -.  ch  ->  et ) )  <-> 
( ( ph  ->  ps )  \/  ( ch  \/  et ) ) )
14 pm4.64 373 . . . . 5  |-  ( ( -.  ta  ->  et ) 
<->  ( ta  \/  et ) )
1514orbi2i 521 . . . 4  |-  ( ( ( -.  ps  ->  ph )  \/  ( -. 
ta  ->  et ) )  <-> 
( ( -.  ps  ->  ph )  \/  ( ta  \/  et ) ) )
1613, 15anbi12i 701 . . 3  |-  ( ( ( ( ph  ->  ps )  \/  ( -. 
ch  ->  et ) )  /\  ( ( -. 
ps  ->  ph )  \/  ( -.  ta  ->  et )
) )  <->  ( (
( ph  ->  ps )  \/  ( ch  \/  et ) )  /\  (
( -.  ps  ->  ph )  \/  ( ta  \/  et ) ) ) )
1711, 16anbi12i 701 . 2  |-  ( ( ( ( ( ph  ->  -.  ps )  \/  ( -.  ch  ->  th ) )  /\  (
( ps  ->  ph )  \/  ( -.  ta  ->  th ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( -.  ch  ->  et )
)  /\  ( ( -.  ps  ->  ph )  \/  ( -.  ta  ->  et ) ) ) )  <-> 
( ( ( (
ph  ->  -.  ps )  \/  ( ch  \/  th ) )  /\  (
( ps  ->  ph )  \/  ( ta  \/  th ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  \/  et ) )  /\  ( ( -. 
ps  ->  ph )  \/  ( ta  \/  et ) ) ) ) )
186, 17bitri 252 1  |-  ( (if- ( ph ,  ch ,  ta )  \/ if- ( ps ,  th ,  et ) )  <->  ( (
( ( ph  ->  -. 
ps )  \/  ( ch  \/  th ) )  /\  ( ( ps 
->  ph )  \/  ( ta  \/  th ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  \/  et ) )  /\  ( ( -.  ps  ->  ph )  \/  ( ta  \/  et ) ) ) ) )
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: (None)
  Copyright terms: Public domain W3C validator