Theorem tbwlem3 1632

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 1627	. . 3 |- ( ( ( ph -> F. ) -> ph ) -> ph )
2		tbw-ax2 1626	. . . 4 |- ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ps ) -> ( ( ( ph -> F. ) -> ph ) -> ph ) ) )
3		tbw-ax1 1625	. . . 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 1629	. . 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 1625	. . 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 1627	. . 3 |- ( ( ( ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ps ) -> ps ) -> ps ) -> ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ps ) -> ps ) ) -> ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ps ) -> ps ) )
8	6, 7	tbwsyl 1629	. 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 1488

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

This theorem is referenced by:  tbwlem4  1633