Theorem tbwlem2 1471

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tbwlem2	|- ((ph -> (ps -> F. )) -> (((ph -> ch) -> th) -> (ps -> th)))

Proof of Theorem tbwlem2

Step	Hyp	Ref	Expression
1		tbw-ax4 1468	. . . . 5 |- ( F. -> ch)
2		tbw-ax1 1465	. . . . . 6 |- ((ps -> F. ) -> (( F. -> ch) -> (ps -> ch)))
3		tbwlem1 1470	. . . . . 6 |- (((ps -> F. ) -> (( F. -> ch) -> (ps -> ch))) -> (( F. -> ch) -> ((ps -> F. ) -> (ps -> ch))))
4	2, 3	ax-mp 5	. . . . 5 |- (( F. -> ch) -> ((ps -> F. ) -> (ps -> ch)))
5	1, 4	ax-mp 5	. . . 4 |- ((ps -> F. ) -> (ps -> ch))
6		tbwlem1 1470	. . . 4 |- (((ps -> F. ) -> (ps -> ch)) -> (ps -> ((ps -> F. ) -> ch)))
7	5, 6	ax-mp 5	. . 3 |- (ps -> ((ps -> F. ) -> ch))
8		tbw-ax1 1465	. . 3 |- ((ph -> (ps -> F. )) -> (((ps -> F. ) -> ch) -> (ph -> ch)))
9		tbw-ax1 1465	. . 3 |- ((ps -> ((ps -> F. ) -> ch)) -> ((((ps -> F. ) -> ch) -> (ph -> ch)) -> (ps -> (ph -> ch))))
10	7, 8, 9	mpsyl 59	. 2 |- ((ph -> (ps -> F. )) -> (ps -> (ph -> ch)))
11		tbw-ax1 1465	. 2 |- ((ps -> (ph -> ch)) -> (((ph -> ch) -> th) -> (ps -> th)))
12	10, 11	tbwsyl 1469	1 |- ((ph -> (ps -> F. )) -> (((ph -> ch) -> th) -> (ps -> th)))

Syntax hints:   -> 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:  tbwlem4  1473