rblem4 - Metamath Proof Explorer

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Theorem rblem4 1685

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	$\/$
rblem4.2	$\/$
rblem4.3	$\/$

**Assertion**
Ref	Expression
rblem4	$\/$ $\/$ $\/$ $\/$ $\/$

**Proof of Theorem rblem4**
Step	Hyp	Ref	Expression
1		rblem4.3	. . . 4 $\/$
2		rblem4.2	. . . 4 $\/$
3	1, 2	rblem1 1682	. . 3 $\/$ $\/$ $\/$
4		rblem4.1	. . 3 $\/$
5	3, 4	rblem1 1682	. 2 $\/$ $\/$ $\/$ $\/$ $\/$
6		rb-ax2 1678	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$
7		rb-ax2 1678	. . . . . 6 $\/$ $\/$ $\/$
8		rb-ax1 1677	. . . . . 6 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
9	7, 8	anmp 1676	. . . . 5 $\/$ $\/$ $\/$ $\/$ $\/$
10		rb-ax2 1678	. . . . 5 $\/$ $\/$ $\/$ $\/$ $\/$
11	9, 10	rbsyl 1681	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$
12	6, 11	rbsyl 1681	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$
13		rb-ax4 1680	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
14		rb-ax2 1678	. . . . . 6 $\/$ $\/$ $\/$ $\/$ $\/$
15		rblem2 1683	. . . . . 6 $\/$ $\/$ $\/$ $\/$
16	14, 15	rbsyl 1681	. . . . 5 $\/$ $\/$ $\/$ $\/$
17		rb-ax3 1679	. . . . . 6 $\/$ $\/$
18		rblem2 1683	. . . . . 6 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
19	17, 18	anmp 1676	. . . . 5 $\/$ $\/$ $\/$
20	16, 19	rblem1 1682	. . . 4 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
21	13, 20	rbsyl 1681	. . 3 $\/$ $\/$ $\/$ $\/$ $\/$
22	12, 21	rbsyl 1681	. 2 $\/$ $\/$ $\/$ $\/$ $\/$
23	5, 22	rbsyl 1681	1 $\/$ $\/$ $\/$ $\/$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3 $\/$ wo 383

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 197 df-or 385 df-an 386

This theorem is referenced by: re2luk1 1690

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