Theorem merco1lem13 1494

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
merco1lem13	|- ((((ph -> ps) -> (ch -> ps)) -> ta) -> (ph -> ta))

Proof of Theorem merco1lem13

Step	Hyp	Ref	Expression
1		merco1 1478	. . . 4 |- (((((ps -> ph) -> (ch -> F. )) -> ph) -> ph) -> ((ph -> ps) -> (ch -> ps)))
2		merco1lem4 1484	. . . 4 |- ((((((ps -> ph) -> (ch -> F. )) -> ph) -> ph) -> ((ph -> ps) -> (ch -> ps))) -> (ph -> ((ph -> ps) -> (ch -> ps))))
3	1, 2	ax-mp 5	. . 3 |- (ph -> ((ph -> ps) -> (ch -> ps)))
4		merco1lem12 1493	. . 3 |- ((ph -> ((ph -> ps) -> (ch -> ps))) -> ((((ta -> ph) -> (ph -> F. )) -> ph) -> ((ph -> ps) -> (ch -> ps))))
5	3, 4	ax-mp 5	. 2 |- ((((ta -> ph) -> (ph -> F. )) -> ph) -> ((ph -> ps) -> (ch -> ps)))
6		merco1 1478	. 2 |- (((((ta -> ph) -> (ph -> F. )) -> ph) -> ((ph -> ps) -> (ch -> ps))) -> ((((ph -> ps) -> (ch -> ps)) -> ta) -> (ph -> ta)))
7	5, 6	ax-mp 5	1 |- ((((ph -> ps) -> (ch -> ps)) -> ta) -> (ph -> ta))

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:  merco1lem14  1495  merco1lem15  1496  retbwax1  1500