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Theorem bilukdc 1328
Description: Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
bilukdc  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ( ph  <->  ps )  <->  ( ( ch  <->  ps )  <->  (
ph 
<->  ch ) ) ) )

Proof of Theorem bilukdc
StepHypRef Expression
1 bicom 138 . . . . . 6  |-  ( (
ph 
<->  ps )  <->  ( ps  <->  ph ) )
21bibi1i 226 . . . . 5  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ( ps  <->  ph )  <->  ch ) )
3 biassdc 1327 . . . . . 6  |-  (DECID  ps  ->  (DECID  ph  ->  (DECID  ch  ->  ( (
( ps  <->  ph )  <->  ch )  <->  ( ps  <->  ( ph  <->  ch )
) ) ) ) )
43imp31 252 . . . . 5  |-  ( ( (DECID  ps  /\ DECID  ph )  /\ DECID  ch )  ->  ( ( ( ps  <->  ph )  <->  ch )  <->  ( ps  <->  (
ph 
<->  ch ) ) ) )
52, 4syl5bb 190 . . . 4  |-  ( ( (DECID  ps  /\ DECID  ph )  /\ DECID  ch )  ->  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ps  <->  (
ph 
<->  ch ) ) ) )
65ancom1s 534 . . 3  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ps  <->  (
ph 
<->  ch ) ) ) )
7 dcbi 878 . . . . . 6  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  <->  ps ) ) )
87imp 122 . . . . 5  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  <->  ps ) )
98adantr 270 . . . 4  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  -> DECID  (
ph 
<->  ps ) )
10 simpr 108 . . . 4  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  -> DECID  ch )
11 dcbi 878 . . . . . 6  |-  (DECID  ph  ->  (DECID  ch 
-> DECID  ( ph  <->  ch ) ) )
12 dcbi 878 . . . . . 6  |-  (DECID  ps  ->  (DECID  (
ph 
<->  ch )  -> DECID  ( ps  <->  ( ph  <->  ch ) ) ) )
1311, 12syl9 71 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  (DECID  ch  -> DECID  ( ps  <->  ( ph  <->  ch ) ) ) ) )
1413imp31 252 . . . 4  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  -> DECID  ( ps  <->  ( ph  <->  ch )
) )
15 biassdc 1327 . . . 4  |-  (DECID  ( ph  <->  ps )  ->  (DECID  ch  ->  (DECID  ( ps  <->  ( ph  <->  ch )
)  ->  ( (
( ( ph  <->  ps )  <->  ch )  <->  ( ps  <->  ( ph  <->  ch ) ) )  <->  ( ( ph 
<->  ps )  <->  ( ch  <->  ( ps  <->  ( ph  <->  ch )
) ) ) ) ) ) )
169, 10, 14, 15syl3c 62 . . 3  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ( ( (
ph 
<->  ps )  <->  ch )  <->  ( ps  <->  ( ph  <->  ch )
) )  <->  ( ( ph 
<->  ps )  <->  ( ch  <->  ( ps  <->  ( ph  <->  ch )
) ) ) ) )
176, 16mpbid 145 . 2  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ( ph  <->  ps )  <->  ( ch  <->  ( ps  <->  ( ph  <->  ch ) ) ) ) )
18 simplr 497 . . 3  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  -> DECID  ps )
1911imp 122 . . . 4  |-  ( (DECID  ph  /\ DECID  ch )  -> DECID 
( ph  <->  ch ) )
2019adantlr 461 . . 3  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  -> DECID  (
ph 
<->  ch ) )
21 biassdc 1327 . . 3  |-  (DECID  ch  ->  (DECID  ps 
->  (DECID  ( ph  <->  ch )  ->  ( ( ( ch  <->  ps )  <->  ( ph  <->  ch )
)  <->  ( ch  <->  ( ps  <->  (
ph 
<->  ch ) ) ) ) ) ) )
2210, 18, 20, 21syl3c 62 . 2  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ( ( ch  <->  ps )  <->  ( ph  <->  ch )
)  <->  ( ch  <->  ( ps  <->  (
ph 
<->  ch ) ) ) ) )
2317, 22bitr4d 189 1  |-  ( ( (DECID 
ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ( ph  <->  ps )  <->  ( ( ch  <->  ps )  <->  (
ph 
<->  ch ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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