Theorem merco1lem11 1492

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem11	|- ((ph -> ps) -> (((ch -> (ph -> ta)) -> F. ) -> ps))

Proof of Theorem merco1lem11

Step	Hyp	Ref	Expression
1		merco1lem5 1485	. . . . . 6 |- ((((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. ) -> F. ) -> (((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ))
2		merco1lem3 1483	. . . . . 6 |- (((((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. ) -> F. ) -> (((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. )) -> (((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )))
3	1, 2	ax-mp 5	. . . . 5 |- (((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. ))
4		merco1lem4 1484	. . . . 5 |- ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )) -> ((((ch -> (ph -> ta)) -> F. ) -> F. ) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )))
5	3, 4	ax-mp 5	. . . 4 |- ((((ch -> (ph -> ta)) -> F. ) -> F. ) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. ))
6		merco1lem5 1485	. . . 4 |- (((((ch -> (ph -> ta)) -> F. ) -> F. ) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )) -> ((ch -> (ph -> ta)) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )))
7	5, 6	ax-mp 5	. . 3 |- ((ch -> (ph -> ta)) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. ))
8		merco1lem4 1484	. . 3 |- (((ch -> (ph -> ta)) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )) -> ((ph -> ta) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )))
9	7, 8	ax-mp 5	. 2 |- ((ph -> ta) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. ))
10		merco1 1478	. . 3 |- (((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> ph) -> ((ph -> ps) -> (((ch -> (ph -> ta)) -> F. ) -> ps)))
11		merco1lem2 1482	. . 3 |- ((((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> ph) -> ((ph -> ps) -> (((ch -> (ph -> ta)) -> F. ) -> ps))) -> (((ph -> ta) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )) -> ((ph -> ps) -> (((ch -> (ph -> ta)) -> F. ) -> ps))))
12	10, 11	ax-mp 5	. 2 |- (((ph -> ta) -> ((((ps -> ph) -> (((ch -> (ph -> ta)) -> F. ) -> F. )) -> F. ) -> F. )) -> ((ph -> ps) -> (((ch -> (ph -> ta)) -> F. ) -> ps)))
13	9, 12	ax-mp 5	1 |- ((ph -> ps) -> (((ch -> (ph -> ta)) -> F. ) -> ps))

Syntax hints:   -> wi 4   F. wfal 1317

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 177  df-tru 1319  df-fal 1320

This theorem is referenced by:  merco1lem12  1493  merco1lem16  1497  merco1lem17  1498