Theorem mercolem3 1504

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
mercolem3	|- ((ps -> ch) -> (ps -> (ph -> ch)))

Proof of Theorem mercolem3

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 |- (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))
2		merco2 1501	. . . 4 |- (((ch -> ph) -> (( F. -> ph) -> ps)) -> ((ps -> ch) -> (ps -> (ph -> ch))))
3		mercolem2 1503	. . . . . . 7 |- (((ps -> (ph -> ch)) -> ps) -> (( F. -> ph) -> (( F. -> ph) -> ps)))
4		merco2 1501	. . . . . . 7 |- ((((ps -> (ph -> ch)) -> ps) -> (( F. -> ph) -> (( F. -> ph) -> ps))) -> (((( F. -> ph) -> ps) -> (ps -> (ph -> ch))) -> (( F. -> ph) -> ((ps -> ch) -> (ps -> (ph -> ch))))))
5	3, 4	ax-mp 5	. . . . . 6 |- (((( F. -> ph) -> ps) -> (ps -> (ph -> ch))) -> (( F. -> ph) -> ((ps -> ch) -> (ps -> (ph -> ch)))))
6		merco2 1501	. . . . . 6 |- ((((( F. -> ph) -> ps) -> (ps -> (ph -> ch))) -> (( F. -> ph) -> ((ps -> ch) -> (ps -> (ph -> ch))))) -> ((((ps -> ch) -> (ps -> (ph -> ch))) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ((ch -> ph) -> (( F. -> ph) -> ps)))))
7	5, 6	ax-mp 5	. . . . 5 |- ((((ps -> ch) -> (ps -> (ph -> ch))) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ((ch -> ph) -> (( F. -> ph) -> ps))))
8		merco2 1501	. . . . 5 |- (((((ps -> ch) -> (ps -> (ph -> ch))) -> (( F. -> ph) -> ps)) -> (( F. -> ph) -> ((ch -> ph) -> (( F. -> ph) -> ps)))) -> ((((ch -> ph) -> (( F. -> ph) -> ps)) -> ((ps -> ch) -> (ps -> (ph -> ch)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ps -> ch) -> (ps -> (ph -> ch)))))))
9	7, 8	ax-mp 5	. . . 4 |- ((((ch -> ph) -> (( F. -> ph) -> ps)) -> ((ps -> ch) -> (ps -> (ph -> ch)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ps -> ch) -> (ps -> (ph -> ch))))))
10	2, 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)))) -> ((ps -> ch) -> (ps -> (ph -> ch)))))
11	1, 10	ax-mp 5	. 2 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ps -> ch) -> (ps -> (ph -> ch))))
12	1, 11	ax-mp 5	1 |- ((ps -> 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:  mercolem4  1505  mercolem7  1508  mercolem8  1509  re1tbw1  1510  re1tbw4  1513