Theorem rblem4 1525

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

Step	Hyp	Ref	Expression
1		rblem4.3	. . . 4 |- (-. ch \/ et)
2		rblem4.2	. . . 4 |- (-. ps \/ ta)
3	1, 2	rblem1 1522	. . 3 |- (-. (ch \/ ps) \/ (et \/ ta))
4		rblem4.1	. . 3 |- (-. ph \/ th)
5	3, 4	rblem1 1522	. 2 |- (-. ((ch \/ ps) \/ ph) \/ ((et \/ ta) \/ th))
6		rb-ax2 1518	. . . 4 |- (-. (ph \/ (ch \/ ps)) \/ ((ch \/ ps) \/ ph))
7		rb-ax2 1518	. . . . . 6 |- (-. (ps \/ ch) \/ (ch \/ ps))
8		rb-ax1 1517	. . . . . 6 |- (-. (-. (ps \/ ch) \/ (ch \/ ps)) \/ (-. (ph \/ (ps \/ ch)) \/ (ph \/ (ch \/ ps))))
9	7, 8	anmp 1516	. . . . 5 |- (-. (ph \/ (ps \/ ch)) \/ (ph \/ (ch \/ ps)))
10		rb-ax2 1518	. . . . 5 |- (-. ((ps \/ ch) \/ ph) \/ (ph \/ (ps \/ ch)))
11	9, 10	rbsyl 1521	. . . 4 |- (-. ((ps \/ ch) \/ ph) \/ (ph \/ (ch \/ ps)))
12	6, 11	rbsyl 1521	. . 3 |- (-. ((ps \/ ch) \/ ph) \/ ((ch \/ ps) \/ ph))
13		rb-ax4 1520	. . . 4 |- (-. (((ps \/ ch) \/ ph) \/ ((ps \/ ch) \/ ph)) \/ ((ps \/ ch) \/ ph))
14		rb-ax2 1518	. . . . . 6 |- (-. (ph \/ (ps \/ ch)) \/ ((ps \/ ch) \/ ph))
15		rblem2 1523	. . . . . 6 |- (-. (ph \/ ps) \/ (ph \/ (ps \/ ch)))
16	14, 15	rbsyl 1521	. . . . 5 |- (-. (ph \/ ps) \/ ((ps \/ ch) \/ ph))
17		rb-ax3 1519	. . . . . 6 |- (-. ch \/ (ps \/ ch))
18		rblem2 1523	. . . . . 6 |- (-. (-. ch \/ (ps \/ ch)) \/ (-. ch \/ ((ps \/ ch) \/ ph)))
19	17, 18	anmp 1516	. . . . 5 |- (-. ch \/ ((ps \/ ch) \/ ph))
20	16, 19	rblem1 1522	. . . 4 |- (-. ((ph \/ ps) \/ ch) \/ (((ps \/ ch) \/ ph) \/ ((ps \/ ch) \/ ph)))
21	13, 20	rbsyl 1521	. . 3 |- (-. ((ph \/ ps) \/ ch) \/ ((ps \/ ch) \/ ph))
22	12, 21	rbsyl 1521	. 2 |- (-. ((ph \/ ps) \/ ch) \/ ((ch \/ ps) \/ ph))
23	5, 22	rbsyl 1521	1 |- (-. ((ph \/ ps) \/ ch) \/ ((et \/ ta) \/ th))

Syntax hints:  -. wn 3   \/ wo 357

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 177  df-or 359  df-an 360

This theorem is referenced by:  re2luk1  1530