Theorem re1luk3 1545

Description: luk-3 1490 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1543 and re1luk2 1544 proves that tbw-ax1 1533, tbw-ax2 1534, tbw-ax3 1535, and tbw-ax4 1536, with ax-mp 5 can be used as a complete axiom system for all of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1luk3

Proof of Theorem re1luk3

Step	Hyp	Ref	Expression
1		tbw-negdf 1532	. . 3
2		tbwlem5 1542	. . 3
3	1, 2	ax-mp 5	. 2
4		tbw-ax4 1536	. . . 4
5		tbw-ax1 1533	. . . . 5
6		tbwlem1 1538	. . . . 5
7	5, 6	ax-mp 5	. . . 4
8	4, 7	ax-mp 5	. . 3
9		tbwlem1 1538	. . 3
10	8, 9	ax-mp 5	. 2
11		tbw-ax1 1533	. 2
12	3, 10, 11	mpsyl 63	1

Syntax hints:  -.wn 3  ->wi 4  wfal 1400

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 185  df-tru 1398  df-fal 1401