Theorem re1luk3 1477

Description: luk-3 1422 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1475 and re1luk2 1476 proves that tbw-ax1 1465, tbw-ax2 1466, tbw-ax3 1467, and tbw-ax4 1468, 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 1464	. . 3 |- (((-. ph -> (ph -> F. )) -> (((ph -> F. ) -> -. ph) -> F. )) -> F. )
2		tbwlem5 1474	. . 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 1468	. . . 4 |- ( F. -> ps)
5		tbw-ax1 1465	. . . . 5 |- ((ph -> F. ) -> (( F. -> ps) -> (ph -> ps)))
6		tbwlem1 1470	. . . . 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 1470	. . 3 |- (((ph -> F. ) -> (ph -> ps)) -> (ph -> ((ph -> F. ) -> ps)))
10	8, 9	ax-mp 5	. 2 |- (ph -> ((ph -> F. ) -> ps))
11		tbw-ax1 1465	. 2 |- ((-. ph -> (ph -> F. )) -> (((ph -> F. ) -> ps) -> (-. ph -> ps)))
12	3, 10, 11	mpsyl 59	1 |- (ph -> (-. ph -> ps))

Syntax hints:  -. wn 3   -> wi 4   F. wfal 1317

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-tru 1319  df-fal 1320

This theorem is referenced by: (None)