Theorem mercolem7 1508

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
mercolem7	|- ((ph -> ps) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps)))

Proof of Theorem mercolem7

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 |- (((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph))))
2		mercolem3 1504	. . . 4 |- (((ph -> ch) -> (th -> ps)) -> ((ph -> ch) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps))))
3		mercolem6 1507	. . . 4 |- ((((ph -> ch) -> (th -> ps)) -> ((ph -> ch) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps)))) -> ((ph -> ch) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps))))
4	2, 3	ax-mp 5	. . 3 |- ((ph -> ch) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps)))
5		mercolem5 1506	. . . 4 |- (ph -> ((ph -> ps) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps))))
6		mercolem4 1505	. . . 4 |- ((ph -> ((ph -> ps) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps)))) -> (((ph -> ch) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> ps) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps))))))
7	5, 6	ax-mp 5	. . 3 |- (((ph -> ch) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps))) -> ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> ps) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps)))))
8	4, 7	ax-mp 5	. 2 |- ((((ph -> ph) -> (( F. -> ph) -> ph)) -> ((ph -> ph) -> (ph -> (ph -> ph)))) -> ((ph -> ps) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps))))
9	1, 8	ax-mp 5	1 |- ((ph -> ps) -> (((ph -> ch) -> (th -> ps)) -> (th -> ps)))

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:  mercolem8  1509