Theorem merlem3 1409

Description: Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merlem3	|- (((ps -> ch) -> ph) -> (ch -> ph))

Proof of Theorem merlem3

Step	Hyp	Ref	Expression
1		merlem2 1408	. . . 4 |- (((-. ch -> -. ch) -> (-. ch -> -. ch)) -> ((ph -> ph) -> (-. ch -> -. ch)))
2		merlem2 1408	. . . 4 |- ((((-. ch -> -. ch) -> (-. ch -> -. ch)) -> ((ph -> ph) -> (-. ch -> -. ch))) -> ((((ch -> ph) -> (-. ps -> -. ps)) -> ps) -> ((ph -> ph) -> (-. ch -> -. ch))))
3	1, 2	ax-mp 5	. . 3 |- ((((ch -> ph) -> (-. ps -> -. ps)) -> ps) -> ((ph -> ph) -> (-. ch -> -. ch)))
4		ax-meredith 1406	. . 3 |- (((((ch -> ph) -> (-. ps -> -. ps)) -> ps) -> ((ph -> ph) -> (-. ch -> -. ch))) -> ((((ph -> ph) -> (-. ch -> -. ch)) -> ch) -> (ps -> ch)))
5	3, 4	ax-mp 5	. 2 |- ((((ph -> ph) -> (-. ch -> -. ch)) -> ch) -> (ps -> ch))
6		ax-meredith 1406	. 2 |- (((((ph -> ph) -> (-. ch -> -. ch)) -> ch) -> (ps -> ch)) -> (((ps -> ch) -> ph) -> (ch -> ph)))
7	5, 6	ax-mp 5	1 |- (((ps -> ch) -> ph) -> (ch -> ph))

Syntax hints:  -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-meredith 1406

This theorem is referenced by:  merlem4  1410  merlem6  1412