Theorem merlem10 1416

Description: Step 19 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
merlem10	|- ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))

Proof of Theorem merlem10

Step	Hyp	Ref	Expression
1		ax-meredith 1406	. 2 |- (((((ph -> ph) -> (-. ph -> -. ph)) -> ph) -> ph) -> ((ph -> ph) -> (ph -> ph)))
2		ax-meredith 1406	. . 3 |- ((((((ph -> ps) -> ph) -> (-. ph -> -. th)) -> ph) -> ph) -> ((ph -> (ph -> ps)) -> (th -> (ph -> ps))))
3		merlem9 1415	. . 3 |- (((((((ph -> ps) -> ph) -> (-. ph -> -. th)) -> ph) -> ph) -> ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))) -> ((((((ph -> ph) -> (-. ph -> -. ph)) -> ph) -> ph) -> ((ph -> ph) -> (ph -> ph))) -> ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))))
4	2, 3	ax-mp 5	. 2 |- ((((((ph -> ph) -> (-. ph -> -. ph)) -> ph) -> ph) -> ((ph -> ph) -> (ph -> ph))) -> ((ph -> (ph -> ps)) -> (th -> (ph -> ps))))
5	1, 4	ax-mp 5	1 |- ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))

Syntax hints:  -. wn 3   -> wi 4

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

This theorem is referenced by:  merlem11  1417