Theorem re1luk3 1637

Description: luk-3 1582 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1635 and re1luk2 1636 proves that tbw-ax1 1625, tbw-ax2 1626, tbw-ax3 1627, and tbw-ax4 1628, 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	|- ( ph -> ( -. ph -> ps ) )

Proof of Theorem re1luk3

Step	Hyp	Ref	Expression
1		tbw-negdf 1624	. . 3 |- ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. )
2		tbwlem5 1634	. . 3 |- ( ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. ) -> ( -. ph -> ( ph -> F. ) ) )
3	1, 2	ax-mp 5	. 2 |- ( -. ph -> ( ph -> F. ) )
4		tbw-ax4 1628	. . . 4 |- ( F. -> ps )
5		tbw-ax1 1625	. . . . 5 |- ( ( ph -> F. ) -> ( ( F. -> ps ) -> ( ph -> ps ) ) )
6		tbwlem1 1630	. . . . 5 |- ( ( ( ph -> F. ) -> ( ( F. -> ps ) -> ( ph -> ps ) ) ) -> ( ( F. -> ps ) -> ( ( ph -> F. ) -> ( ph -> ps ) ) ) )
7	5, 6	ax-mp 5	. . . 4 |- ( ( F. -> ps ) -> ( ( ph -> F. ) -> ( ph -> ps ) ) )
8	4, 7	ax-mp 5	. . 3 |- ( ( ph -> F. ) -> ( ph -> ps ) )
9		tbwlem1 1630	. . 3 |- ( ( ( ph -> F. ) -> ( ph -> ps ) ) -> ( ph -> ( ( ph -> F. ) -> ps ) ) )
10	8, 9	ax-mp 5	. 2 |- ( ph -> ( ( ph -> F. ) -> ps ) )
11		tbw-ax1 1625	. 2 |- ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> ps ) -> ( -. ph -> ps ) ) )
12	3, 10, 11	mpsyl 68	1 |- ( ph -> ( -. ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   F. wfal 1488

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-tru 1486  df-fal 1489

This theorem is referenced by: (None)