Theorem merco1lem3 1643

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
merco1lem3	|- ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) )

Proof of Theorem merco1lem3

Step	Hyp	Ref	Expression
1		merco1lem2 1642	. . 3 |- ( ( ( ph -> ph ) -> F. ) -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> F. ) )
2		retbwax2 1641	. . . 4 |- ( ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) -> ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) )
3		merco1lem2 1642	. . . 4 |- ( ( ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) -> ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) ) -> ( ( ( ( ph -> ph ) -> F. ) -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> F. ) ) -> ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) ) )
4	2, 3	ax-mp 5	. . 3 |- ( ( ( ( ph -> ph ) -> F. ) -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> F. ) ) -> ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) )
5	1, 4	ax-mp 5	. 2 |- ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) )
6		merco1lem2 1642	. . 3 |- ( ( ( ch -> ph ) -> F. ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> F. ) )
7		retbwax2 1641	. . . 4 |- ( ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) -> ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) ) )
8		merco1lem2 1642	. . . 4 |- ( ( ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) -> ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) ) ) -> ( ( ( ( ch -> ph ) -> F. ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> F. ) ) -> ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) ) ) )
9	7, 8	ax-mp 5	. . 3 |- ( ( ( ( ch -> ph ) -> F. ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> F. ) ) -> ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) ) )
10	6, 9	ax-mp 5	. 2 |- ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) )
11	5, 10	ax-mp 5	1 |- ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) )

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:  merco1lem4  1644  merco1lem6  1646  merco1lem11  1652  merco1lem12  1653  merco1lem18  1659