Theorem mercolem4 1505

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mercolem4	|- ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))

Proof of Theorem mercolem4

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 |- (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))
2		merco2 1501	. . . 4 |- ((((et -> ph) -> ph) -> (( F. -> ph) -> th)) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))))
3		merco2 1501	. . . . . . . . 9 |- (((ph -> ph) -> (( F. -> ph) -> (th -> ch))) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))
4		mercolem1 1502	. . . . . . . . 9 |- ((((ph -> ph) -> (( F. -> ph) -> (th -> ch))) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))) -> ((( F. -> ph) -> (th -> ch)) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))))
5	3, 4	ax-mp 5	. . . . . . . 8 |- ((( F. -> ph) -> (th -> ch)) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))))
6		mercolem1 1502	. . . . . . . 8 |- (((( F. -> ph) -> (th -> ch)) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))) -> ((th -> ch) -> (( F. -> ph) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))))))
7	5, 6	ax-mp 5	. . . . . . 7 |- ((th -> ch) -> (( F. -> ph) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))))
8		merco2 1501	. . . . . . 7 |- (((th -> ch) -> (( F. -> ph) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))))) -> ((((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))) -> th) -> (((et -> ph) -> ph) -> (( F. -> ph) -> th))))
9	7, 8	ax-mp 5	. . . . . 6 |- ((((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))) -> th) -> (((et -> ph) -> ph) -> (( F. -> ph) -> th)))
10		mercolem3 1504	. . . . . 6 |- (((((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))) -> th) -> (((et -> ph) -> ph) -> (( F. -> ph) -> th))) -> ((((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))) -> th) -> (( F. -> ph) -> (((et -> ph) -> ph) -> (( F. -> ph) -> th)))))
11	9, 10	ax-mp 5	. . . . 5 |- ((((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))) -> th) -> (( F. -> ph) -> (((et -> ph) -> ph) -> (( F. -> ph) -> th))))
12		merco2 1501	. . . . 5 |- (((((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))) -> th) -> (( F. -> ph) -> (((et -> ph) -> ph) -> (( F. -> ph) -> th)))) -> (((((et -> ph) -> ph) -> (( F. -> ph) -> th)) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))))))
13	11, 12	ax-mp 5	. . . 4 |- (((((et -> ph) -> ph) -> (( F. -> ph) -> th)) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))))))
14	2, 13	ax-mp 5	. . 3 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph))))))
15	1, 14	ax-mp 5	. 2 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> ph)))))
16	1, 15	ax-mp 5	1 |- ((th -> (et -> ph)) -> (((th -> ch) -> ph) -> (ta -> (et -> 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:  mercolem6  1507  mercolem7  1508