Theorem merco1lem4 1644

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1638. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem4	|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )

Proof of Theorem merco1lem4

Step	Hyp	Ref	Expression
1		merco1lem3 1643	. . 3 |- ( ( ( ( ps -> F. ) -> ( ph -> F. ) ) -> ( ( ch -> ph ) -> F. ) ) -> ( ( ch -> ph ) -> ( ps -> F. ) ) )
2		merco1 1638	. . 3 |- ( ( ( ( ( ps -> F. ) -> ( ph -> F. ) ) -> ( ( ch -> ph ) -> F. ) ) -> ( ( ch -> ph ) -> ( ps -> F. ) ) ) -> ( ( ( ( ch -> ph ) -> ( ps -> F. ) ) -> ps ) -> ( ph -> ps ) ) )
3	1, 2	ax-mp 5	. 2 |- ( ( ( ( ch -> ph ) -> ( ps -> F. ) ) -> ps ) -> ( ph -> ps ) )
4		merco1 1638	. 2 |- ( ( ( ( ( ch -> ph ) -> ( ps -> F. ) ) -> ps ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) )
5	3, 4	ax-mp 5	1 |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> 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:  merco1lem5  1645  merco1lem11  1652  merco1lem13  1654  merco1lem17  1658  merco1lem18  1659