Theorem tbwlem3

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.)

	Ref	Expression
Assertion	tbwlem3	|- ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ps ) -> ps )

Proof

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