Theorem tbwlem3 1472

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
tbwlem3	|- (((((ph -> F. ) -> ph) -> ph) -> ps) -> ps)

Proof of Theorem tbwlem3

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

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 is referenced by:  tbwlem4  1473