Theorem merco1lem14 1495

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
merco1lem14	|- ((((ph -> ps) -> ps) -> ch) -> (ph -> ch))

Proof of Theorem merco1lem14

Step	Hyp	Ref	Expression
1		merco1lem13 1494	. . . 4 |- ((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps)))
2		merco1lem8 1489	. . . . . 6 |- (((((ph -> ((ph -> ps) -> ps)) -> ph) -> (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> F. )) -> ph) -> (((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)))
3		merco1 1478	. . . . . 6 |- ((((((ph -> ((ph -> ps) -> ps)) -> ph) -> (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> F. )) -> ph) -> (((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps))) -> (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> (ph -> ((ph -> ps) -> ps)))))
4	2, 3	ax-mp 5	. . . . 5 |- (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> (ph -> ((ph -> ps) -> ps))))
5		merco1lem9 1490	. . . . 5 |- ((((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> (ph -> ((ph -> ps) -> ps)))) -> (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> (ph -> ((ph -> ps) -> ps))))
6	4, 5	ax-mp 5	. . . 4 |- (((((ph -> ps) -> ((ph -> ps) -> ps)) -> ((ph -> ps) -> ps)) -> (ph -> ((ph -> ps) -> ps))) -> (ph -> ((ph -> ps) -> ps)))
7	1, 6	ax-mp 5	. . 3 |- (ph -> ((ph -> ps) -> ps))
8		merco1lem12 1493	. . 3 |- ((ph -> ((ph -> ps) -> ps)) -> ((((ch -> ph) -> (ph -> F. )) -> ph) -> ((ph -> ps) -> ps)))
9	7, 8	ax-mp 5	. 2 |- ((((ch -> ph) -> (ph -> F. )) -> ph) -> ((ph -> ps) -> ps))
10		merco1 1478	. 2 |- (((((ch -> ph) -> (ph -> F. )) -> ph) -> ((ph -> ps) -> ps)) -> ((((ph -> ps) -> ps) -> ch) -> (ph -> ch)))
11	9, 10	ax-mp 5	1 |- ((((ph -> ps) -> ps) -> ch) -> (ph -> ch))

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:  merco1lem15  1496  retbwax1  1500