Theorem merco1lem3 1483

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (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 1482	. . 3 |- (((ph -> ph) -> F. ) -> (((ph -> ph) -> (ph -> F. )) -> F. ))
2		retbwax2 1481	. . . 4 |- ((((ph -> ph) -> (ph -> F. )) -> (ph -> ph)) -> (ph -> (((ph -> ph) -> (ph -> F. )) -> (ph -> ph))))
3		merco1lem2 1482	. . . 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 1482	. . 3 |- (((ch -> ph) -> F. ) -> (((ph -> ps) -> (ch -> F. )) -> F. ))
7		retbwax2 1481	. . . 4 |- ((((ph -> ps) -> (ch -> F. )) -> (ch -> ph)) -> ((ph -> (((ph -> ph) -> (ph -> F. )) -> (ph -> ph))) -> (((ph -> ps) -> (ch -> F. )) -> (ch -> ph))))
8		merco1lem2 1482	. . . 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 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:  merco1lem4  1484  merco1lem6  1486  merco1lem11  1492  merco1lem12  1493  merco1lem18  1499