Theorem merco1lem6 1486

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem6	|- ((ph -> (ph -> ps)) -> (ch -> (ph -> ps)))

Proof of Theorem merco1lem6

Step	Hyp	Ref	Expression
1		merco1lem5 1485	. . . . 5 |- (((((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> F. ) -> F. ) -> ((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ))
2		merco1lem3 1483	. . . . 5 |- ((((((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> F. ) -> F. ) -> ((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. )) -> ((((ph -> ps) -> F. ) -> (ch -> F. )) -> (((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> F. )))
3	1, 2	ax-mp 5	. . . 4 |- ((((ph -> ps) -> F. ) -> (ch -> F. )) -> (((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> F. ))
4		merco1lem5 1485	. . . 4 |- (((((ph -> ps) -> F. ) -> (ch -> F. )) -> (((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> F. )) -> ((ph -> ps) -> (((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> F. )))
5	3, 4	ax-mp 5	. . 3 |- ((ph -> ps) -> (((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> F. ))
6		merco1lem3 1483	. . 3 |- (((ph -> ps) -> (((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> F. )) -> (((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> ph))
7	5, 6	ax-mp 5	. 2 |- (((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> ph)
8		merco1 1478	. 2 |- ((((((ph -> ps) -> F. ) -> (ch -> F. )) -> F. ) -> ph) -> ((ph -> (ph -> ps)) -> (ch -> (ph -> ps))))
9	7, 8	ax-mp 5	1 |- ((ph -> (ph -> ps)) -> (ch -> (ph -> 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:  merco1lem7  1487  merco1lem8  1489