Theorem mercolem6 1507

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
mercolem6	|- ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))

Proof of Theorem mercolem6

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 |- (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))
2		mercolem1 1502	. . . . . . . 8 |- (((ph -> (ph -> (ps -> (ph -> ch)))) -> (ph -> ch)) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))
3		mercolem1 1502	. . . . . . . 8 |- ((((ph -> (ph -> (ps -> (ph -> ch)))) -> (ph -> ch)) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))) -> ((ph -> ch) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))
4	2, 3	ax-mp 5	. . . . . . 7 |- ((ph -> ch) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))
5		mercolem5 1506	. . . . . . . 8 |- (ph -> ((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))
6		mercolem4 1505	. . . . . . . 8 |- ((ph -> ((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))) -> (((ph -> ch) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))))
7	5, 6	ax-mp 5	. . . . . . 7 |- (((ph -> ch) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))))
8	4, 7	ax-mp 5	. . . . . 6 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))
9	1, 8	ax-mp 5	. . . . 5 |- ((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))
10		mercolem1 1502	. . . . . . . 8 |- (((ph -> (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))
11		mercolem1 1502	. . . . . . . 8 |- ((((ph -> (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))) -> (((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))))
12	10, 11	ax-mp 5	. . . . . . 7 |- (((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))))
13		mercolem5 1506	. . . . . . . 8 |- ((ph -> (ps -> (ph -> ch))) -> (((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))))
14		mercolem4 1505	. . . . . . . 8 |- (((ph -> (ps -> (ph -> ch))) -> (((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))))) -> ((((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> (((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))))))
15	13, 14	ax-mp 5	. . . . . . 7 |- ((((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> (((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))))))
16	12, 15	ax-mp 5	. . . . . 6 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> (((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))))
17	1, 16	ax-mp 5	. . . . 5 |- (((ph -> (ps -> (ph -> ch))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))))
18	9, 17	ax-mp 5	. . . 4 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))))
19	1, 18	ax-mp 5	. . 3 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch)))))
20	1, 19	ax-mp 5	. 2 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> (ps -> (ph -> ch))) -> (ps -> (ph -> ch))))
21	1, 20	ax-mp 5	1 |- ((ph -> (ps -> (ph -> ch))) -> (ps -> (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:  mercolem7  1508  re1tbw1  1510  re1tbw2  1511  re1tbw3  1512