Theorem mercolem5 1506

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
mercolem5	|- (th -> ((th -> ph) -> (ta -> (ch -> ph))))

Proof of Theorem mercolem5

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 |- (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))
2		merco2 1501	. . . . 5 |- (((ph -> ph) -> (( F. -> ph) -> th)) -> ((th -> ph) -> (ta -> (ch -> ph))))
3		mercolem1 1502	. . . . 5 |- ((((ph -> ph) -> (( F. -> ph) -> th)) -> ((th -> ph) -> (ta -> (ch -> ph)))) -> ((( F. -> ph) -> th) -> (th -> ((th -> ph) -> (ta -> (ch -> ph))))))
4	2, 3	ax-mp 5	. . . 4 |- ((( F. -> ph) -> th) -> (th -> ((th -> ph) -> (ta -> (ch -> ph)))))
5		mercolem2 1503	. . . . 5 |- (((th -> ((th -> ph) -> (ta -> (ch -> ph)))) -> th) -> (( F. -> ph) -> (( F. -> ph) -> th)))
6		merco2 1501	. . . . 5 |- ((((th -> ((th -> ph) -> (ta -> (ch -> ph)))) -> th) -> (( F. -> ph) -> (( F. -> ph) -> th))) -> (((( F. -> ph) -> th) -> (th -> ((th -> ph) -> (ta -> (ch -> ph))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> (th -> ((th -> ph) -> (ta -> (ch -> ph))))))))
7	5, 6	ax-mp 5	. . . 4 |- (((( F. -> ph) -> th) -> (th -> ((th -> ph) -> (ta -> (ch -> ph))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> (th -> ((th -> ph) -> (ta -> (ch -> ph)))))))
8	4, 7	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 -> ((th -> ph) -> (ta -> (ch -> ph))))))
9	1, 8	ax-mp 5	. 2 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> (th -> ((th -> ph) -> (ta -> (ch -> ph)))))
10	1, 9	ax-mp 5	1 |- (th -> ((th -> ph) -> (ta -> (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:  mercolem6  1507  mercolem7  1508