Theorem mercolem8 1509

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
mercolem8	|- ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch)))))

Proof of Theorem mercolem8

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 |- (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))
2		merco2 1501	. . . . 5 |- ((((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch)))))
3		mercolem3 1504	. . . . 5 |- (((((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch))))) -> ((((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)) -> ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch)))))))
4	2, 3	ax-mp 5	. . . 4 |- ((((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)) -> ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch))))))
5		mercolem7 1508	. . . . . 6 |- ((ph -> ps) -> (((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)))
6		mercolem7 1508	. . . . . 6 |- (((ph -> ps) -> (((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps))) -> ((((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch))))) -> (( F. -> ph) -> (((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)))) -> (( F. -> ph) -> (((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)))))
7	5, 6	ax-mp 5	. . . . 5 |- ((((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch))))) -> (( F. -> ph) -> (((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)))) -> (( F. -> ph) -> (((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps))))
8		merco2 1501	. . . . 5 |- (((((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch))))) -> (( F. -> ph) -> (((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)))) -> (( F. -> ph) -> (((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)))) -> (((((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)) -> ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch)))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch)))))))))
9	7, 8	ax-mp 5	. . . 4 |- (((((ph -> ch) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ps)) -> ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch)))))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch))))))))
10	4, 9	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) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch)))))))
11	1, 10	ax-mp 5	. 2 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (ph -> ch))))))
12	1, 11	ax-mp 5	1 |- ((ph -> ps) -> ((ps -> (ph -> ch)) -> (ta -> (th -> (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:  re1tbw1  1510