Theorem merlem1 922

Description: Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.)

Assertion

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

Proof of Theorem merlem1

Step	Hyp	Ref	Expression
1		meredith 921	. . 3 |- (((((-. ph -> ps) -> (-. (-. ta -> -. ch) -> -. -. (-. ph -> ps))) -> (-. ta -> -. ch)) -> ta) -> ((ta -> -. ph) -> (-. (-. ph -> ps) -> -. ph)))
2		meredith 921	. . 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 7	. 2 |- ((((ta -> -. ph) -> (-. (-. ph -> ps) -> -. ph)) -> (-. ph -> ps)) -> (ch -> (-. ph -> ps)))
4		meredith 921	. 2 |- (((((ta -> -. ph) -> (-. (-. ph -> ps) -> -. ph)) -> (-. ph -> ps)) -> (ch -> (-. ph -> ps))) -> (((ch -> (-. ph -> ps)) -> ta) -> (ph -> ta)))
5	3, 4	ax-mp 7	1 |- (((ch -> (-. ph -> ps)) -> ta) -> (ph -> ta))

Syntax hints:  -. wn 2   -> wi 3

This theorem is referenced by:  merlem2 923  merlem5 926  luk-3 937

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

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