Theorem merlem3 924

Description: Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

Assertion

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

Proof of Theorem merlem3

Step	Hyp	Ref	Expression
1		merlem2 923	. . . 4 |- (((-. ch -> -. ch) -> (-. ch -> -. ch)) -> ((ph -> ph) -> (-. ch -> -. ch)))
2		merlem2 923	. . . 4 |- ((((-. ch -> -. ch) -> (-. ch -> -. ch)) -> ((ph -> ph) -> (-. ch -> -. ch))) -> ((((ch -> ph) -> (-. ps -> -. ps)) -> ps) -> ((ph -> ph) -> (-. ch -> -. ch))))
3	1, 2	ax-mp 7	. . 3 |- ((((ch -> ph) -> (-. ps -> -. ps)) -> ps) -> ((ph -> ph) -> (-. ch -> -. ch)))
4		meredith 921	. . 3 |- (((((ch -> ph) -> (-. ps -> -. ps)) -> ps) -> ((ph -> ph) -> (-. ch -> -. ch))) -> ((((ph -> ph) -> (-. ch -> -. ch)) -> ch) -> (ps -> ch)))
5	3, 4	ax-mp 7	. 2 |- ((((ph -> ph) -> (-. ch -> -. ch)) -> ch) -> (ps -> ch))
6		meredith 921	. 2 |- (((((ph -> ph) -> (-. ch -> -. ch)) -> ch) -> (ps -> ch)) -> (((ps -> ch) -> ph) -> (ch -> ph)))
7	5, 6	ax-mp 7	1 |- (((ps -> ch) -> ph) -> (ch -> ph))

Syntax hints:  -. wn 2   -> wi 3

This theorem is referenced by:  merlem4 925  merlem6 927

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

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