Theorem mercolem2 1503

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
mercolem2	|- (((ph -> ps) -> ph) -> (ch -> (th -> ph)))

Proof of Theorem mercolem2

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 |- (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))
2		merco2 1501	. . . 4 |- (((ph -> ph) -> (( F. -> ph) -> (ph -> ps))) -> (((ph -> ps) -> ph) -> (ch -> (th -> ph))))
3		merco2 1501	. . . . . . . 8 |- (((ph -> ps) -> (( F. -> ph) -> F. )) -> (( F. -> ph) -> (ch -> (th -> ph))))
4		merco2 1501	. . . . . . . 8 |- ((((ph -> ps) -> (( F. -> ph) -> F. )) -> (( F. -> ph) -> (ch -> (th -> ph)))) -> (((ch -> (th -> ph)) -> (ph -> ps)) -> (( F. -> ph) -> (( F. -> ph) -> (ph -> ps)))))
5	3, 4	ax-mp 5	. . . . . . 7 |- (((ch -> (th -> ph)) -> (ph -> ps)) -> (( F. -> ph) -> (( F. -> ph) -> (ph -> ps))))
6		merco2 1501	. . . . . . 7 |- ((((ch -> (th -> ph)) -> (ph -> ps)) -> (( F. -> ph) -> (( F. -> ph) -> (ph -> ps)))) -> (((( F. -> ph) -> (ph -> ps)) -> (ch -> (th -> ph))) -> (( F. -> ph) -> (((ph -> ps) -> ph) -> (ch -> (th -> ph))))))
7	5, 6	ax-mp 5	. . . . . 6 |- (((( F. -> ph) -> (ph -> ps)) -> (ch -> (th -> ph))) -> (( F. -> ph) -> (((ph -> ps) -> ph) -> (ch -> (th -> ph)))))
8		merco2 1501	. . . . . 6 |- ((((( F. -> ph) -> (ph -> ps)) -> (ch -> (th -> ph))) -> (( F. -> ph) -> (((ph -> ps) -> ph) -> (ch -> (th -> ph))))) -> (((((ph -> ps) -> ph) -> (ch -> (th -> ph))) -> (( F. -> ph) -> (ph -> ps))) -> (( F. -> ph) -> ((ph -> ph) -> (( F. -> ph) -> (ph -> ps))))))
9	7, 8	ax-mp 5	. . . . 5 |- (((((ph -> ps) -> ph) -> (ch -> (th -> ph))) -> (( F. -> ph) -> (ph -> ps))) -> (( F. -> ph) -> ((ph -> ph) -> (( F. -> ph) -> (ph -> ps)))))
10		merco2 1501	. . . . 5 |- ((((((ph -> ps) -> ph) -> (ch -> (th -> ph))) -> (( F. -> ph) -> (ph -> ps))) -> (( F. -> ph) -> ((ph -> ph) -> (( F. -> ph) -> (ph -> ps))))) -> ((((ph -> ph) -> (( F. -> ph) -> (ph -> ps))) -> (((ph -> ps) -> ph) -> (ch -> (th -> 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 -> (th -> ph)))))))
11	9, 10	ax-mp 5	. . . 4 |- ((((ph -> ph) -> (( F. -> ph) -> (ph -> ps))) -> (((ph -> ps) -> ph) -> (ch -> (th -> 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 -> (th -> ph))))))
12	2, 11	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 -> (th -> ph)))))
13	1, 12	ax-mp 5	. 2 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> (((ph -> ps) -> ph) -> (ch -> (th -> ph))))
14	1, 13	ax-mp 5	1 |- (((ph -> ps) -> ph) -> (ch -> (th -> 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:  mercolem3  1504  mercolem5  1506  re1tbw3  1512