Theorem merco1lem12 1653

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
merco1lem12	|- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ps ) )

Proof of Theorem merco1lem12

Step	Hyp	Ref	Expression
1		merco1lem3 1643	. . . . 5 |- ( ( ( ( ph -> ta ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> ( ch -> F. ) ) -> ( ch -> ( ph -> ta ) ) )
2		merco1 1638	. . . . 5 |- ( ( ( ( ( ph -> ta ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> ( ch -> F. ) ) -> ( ch -> ( ph -> ta ) ) ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) ) )
3	1, 2	ax-mp 5	. . . 4 |- ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) )
4		merco1lem9 1650	. . . 4 |- ( ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) )
5	3, 4	ax-mp 5	. . 3 |- ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph )
6		merco1lem11 1652	. . 3 |- ( ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) -> ( ( ( ( ps -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> F. ) -> ph ) )
7	5, 6	ax-mp 5	. 2 |- ( ( ( ( ps -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> F. ) -> ph )
8		merco1 1638	. 2 |- ( ( ( ( ( ps -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> F. ) -> ph ) -> ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ps ) ) )
9	7, 8	ax-mp 5	1 |- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> 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 depends on definitions:  df-bi 197  df-tru 1486  df-fal 1489

This theorem is referenced by:  merco1lem13  1654  merco1lem14  1655