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Theorem rblem4 1515
Description: Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
rblem4.1  |-  ( -. 
ph  \/  th )
rblem4.2  |-  ( -. 
ps  \/  ta )
rblem4.3  |-  ( -. 
ch  \/  et )
Assertion
Ref Expression
rblem4  |-  ( -.  ( ( ph  \/  ps )  \/  ch )  \/  ( ( et  \/  ta )  \/ 
th ) )

Proof of Theorem rblem4
StepHypRef Expression
1 rblem4.3 . . . 4  |-  ( -. 
ch  \/  et )
2 rblem4.2 . . . 4  |-  ( -. 
ps  \/  ta )
31, 2rblem1 1512 . . 3  |-  ( -.  ( ch  \/  ps )  \/  ( et  \/  ta ) )
4 rblem4.1 . . 3  |-  ( -. 
ph  \/  th )
53, 4rblem1 1512 . 2  |-  ( -.  ( ( ch  \/  ps )  \/  ph )  \/  ( ( et  \/  ta )  \/  th )
)
6 rb-ax2 1508 . . . 4  |-  ( -.  ( ph  \/  ( ch  \/  ps ) )  \/  ( ( ch  \/  ps )  \/ 
ph ) )
7 rb-ax2 1508 . . . . . 6  |-  ( -.  ( ps  \/  ch )  \/  ( ch  \/  ps ) )
8 rb-ax1 1507 . . . . . 6  |-  ( -.  ( -.  ( ps  \/  ch )  \/  ( ch  \/  ps ) )  \/  ( -.  ( ph  \/  ( ps  \/  ch ) )  \/  ( ph  \/  ( ch  \/  ps ) ) ) )
97, 8anmp 1506 . . . . 5  |-  ( -.  ( ph  \/  ( ps  \/  ch ) )  \/  ( ph  \/  ( ch  \/  ps ) ) )
10 rb-ax2 1508 . . . . 5  |-  ( -.  ( ( ps  \/  ch )  \/  ph )  \/  ( ph  \/  ( ps  \/  ch ) ) )
119, 10rbsyl 1511 . . . 4  |-  ( -.  ( ( ps  \/  ch )  \/  ph )  \/  ( ph  \/  ( ch  \/  ps ) ) )
126, 11rbsyl 1511 . . 3  |-  ( -.  ( ( ps  \/  ch )  \/  ph )  \/  ( ( ch  \/  ps )  \/  ph )
)
13 rb-ax4 1510 . . . 4  |-  ( -.  ( ( ( ps  \/  ch )  \/ 
ph )  \/  (
( ps  \/  ch )  \/  ph ) )  \/  ( ( ps  \/  ch )  \/ 
ph ) )
14 rb-ax2 1508 . . . . . 6  |-  ( -.  ( ph  \/  ( ps  \/  ch ) )  \/  ( ( ps  \/  ch )  \/ 
ph ) )
15 rblem2 1513 . . . . . 6  |-  ( -.  ( ph  \/  ps )  \/  ( ph  \/  ( ps  \/  ch ) ) )
1614, 15rbsyl 1511 . . . . 5  |-  ( -.  ( ph  \/  ps )  \/  ( ( ps  \/  ch )  \/ 
ph ) )
17 rb-ax3 1509 . . . . . 6  |-  ( -. 
ch  \/  ( ps  \/  ch ) )
18 rblem2 1513 . . . . . 6  |-  ( -.  ( -.  ch  \/  ( ps  \/  ch ) )  \/  ( -.  ch  \/  ( ( ps  \/  ch )  \/  ph ) ) )
1917, 18anmp 1506 . . . . 5  |-  ( -. 
ch  \/  ( ( ps  \/  ch )  \/ 
ph ) )
2016, 19rblem1 1512 . . . 4  |-  ( -.  ( ( ph  \/  ps )  \/  ch )  \/  ( (
( ps  \/  ch )  \/  ph )  \/  ( ( ps  \/  ch )  \/  ph )
) )
2113, 20rbsyl 1511 . . 3  |-  ( -.  ( ( ph  \/  ps )  \/  ch )  \/  ( ( ps  \/  ch )  \/ 
ph ) )
2212, 21rbsyl 1511 . 2  |-  ( -.  ( ( ph  \/  ps )  \/  ch )  \/  ( ( ch  \/  ps )  \/ 
ph ) )
235, 22rbsyl 1511 1  |-  ( -.  ( ( ph  \/  ps )  \/  ch )  \/  ( ( et  \/  ta )  \/ 
th ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357
This theorem is referenced by:  re2luk1  1520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360
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