Theorem merlem1 1407

Description: Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merlem1	|- (((ch -> (-. ph -> ps)) -> ta) -> (ph -> ta))

Proof of Theorem merlem1

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

Syntax hints:  -. wn 3   -> wi 4

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

This theorem is referenced by:  merlem2  1408  merlem5  1411  luk-3  1422