Theorem merco1lem13 1654

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
merco1lem13	|- ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph -> ta ) )

Proof of Theorem merco1lem13

Step	Hyp	Ref	Expression
1		merco1 1638	. . . 4 |- ( ( ( ( ( ps -> ph ) -> ( ch -> F. ) ) -> ph ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) )
2		merco1lem4 1644	. . . 4 |- ( ( ( ( ( ( ps -> ph ) -> ( ch -> F. ) ) -> ph ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( ph -> ( ( ph -> ps ) -> ( ch -> ps ) ) )
4		merco1lem12 1653	. . 3 |- ( ( ph -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ( ( ( ta -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) )
5	3, 4	ax-mp 5	. 2 |- ( ( ( ( ta -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) )
6		merco1 1638	. 2 |- ( ( ( ( ( ta -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph -> ta ) ) )
7	5, 6	ax-mp 5	1 |- ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph -> ta ) )

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:  merco1lem14  1655  merco1lem15  1656  retbwax1  1660