Theorem tbwlem1 1470

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
tbwlem1	|- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))

Proof of Theorem tbwlem1

Step	Hyp	Ref	Expression
1		tbw-ax2 1466	. . . 4 |- (ps -> ((ps -> ch) -> ps))
2		tbw-ax1 1465	. . . 4 |- (((ps -> ch) -> ps) -> ((ps -> ch) -> ((ps -> ch) -> ch)))
3	1, 2	tbwsyl 1469	. . 3 |- (ps -> ((ps -> ch) -> ((ps -> ch) -> ch)))
4		tbw-ax1 1465	. . . 4 |- (((ps -> ch) -> ((ps -> ch) -> ch)) -> ((((ps -> ch) -> ch) -> ch) -> ((ps -> ch) -> ch)))
5		tbw-ax3 1467	. . . 4 |- (((((ps -> ch) -> ch) -> ch) -> ((ps -> ch) -> ch)) -> ((ps -> ch) -> ch))
6	4, 5	tbwsyl 1469	. . 3 |- (((ps -> ch) -> ((ps -> ch) -> ch)) -> ((ps -> ch) -> ch))
7	3, 6	tbwsyl 1469	. 2 |- (ps -> ((ps -> ch) -> ch))
8		tbw-ax1 1465	. 2 |- ((ph -> (ps -> ch)) -> (((ps -> ch) -> ch) -> (ph -> ch)))
9		tbw-ax1 1465	. 2 |- ((ps -> ((ps -> ch) -> ch)) -> ((((ps -> ch) -> ch) -> (ph -> ch)) -> (ps -> (ph -> ch))))
10	7, 8, 9	mpsyl 59	1 |- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  tbwlem2  1471  tbwlem4  1473  tbwlem5  1474  re1luk3  1477