Theorem merco1lem14 1655

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1638. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem14	|- ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) )

Proof of Theorem merco1lem14

Step	Hyp	Ref	Expression
1		merco1lem13 1654	. . . 4 |- ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) )
2		merco1lem8 1649	. . . . . 6 |- ( ( ( ( ( ph -> ( ( ph -> ps ) -> ps ) ) -> ph ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> F. ) ) -> ph ) -> ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) )
3		merco1 1638	. . . . . 6 |- ( ( ( ( ( ( ph -> ( ( ph -> ps ) -> ps ) ) -> ph ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> F. ) ) -> ph ) -> ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) ) )
4	2, 3	ax-mp 5	. . . . 5 |- ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) )
5		merco1lem9 1650	. . . . 5 |- ( ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) )
6	4, 5	ax-mp 5	. . . 4 |- ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) )
7	1, 6	ax-mp 5	. . 3 |- ( ph -> ( ( ph -> ps ) -> ps ) )
8		merco1lem12 1653	. . 3 |- ( ( ph -> ( ( ph -> ps ) -> ps ) ) -> ( ( ( ( ch -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ps ) ) )
9	7, 8	ax-mp 5	. 2 |- ( ( ( ( ch -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ps ) )
10		merco1 1638	. 2 |- ( ( ( ( ( ch -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) ) )
11	9, 10	ax-mp 5	1 |- ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) )

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 depends on definitions:  df-bi 197  df-tru 1486  df-fal 1489

This theorem is referenced by:  merco1lem15  1656  retbwax1  1660