Theorem merco1lem12 1493

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

Proof of Theorem merco1lem12

Step	Hyp	Ref	Expression
1		merco1lem3 1483	. . . . 5 |- ((((ph -> ta) -> (((ch -> (ph -> ta)) -> ph) -> F. )) -> (ch -> F. )) -> (ch -> (ph -> ta)))
2		merco1 1478	. . . . 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 1490	. . . 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 1492	. . 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 1478	. 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 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:  merco1lem13  1494  merco1lem14  1495