Theorem merlem7 1055

Description: Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

Assertion

Ref	Expression
merlem7	|- (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))

Proof of Theorem merlem7

Step	Hyp	Ref	Expression
1		merlem4 1052	. 2 |- ((ps -> ch) -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))
2		merlem6 1054	. . . 4 |- ((((ch -> ta) -> (-. th -> -. ps)) -> th) -> (((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph)))
3		meredith 1047	. . . 4 |- (((((ch -> ta) -> (-. th -> -. ps)) -> th) -> (((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph))) -> (((((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph)) -> ch) -> (ps -> ch)))
4	2, 3	ax-mp 7	. . 3 |- (((((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph)) -> ch) -> (ps -> ch))
5		meredith 1047	. . 3 |- ((((((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph)) -> ch) -> (ps -> ch)) -> (((ps -> ch) -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))) -> (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))))
6	4, 5	ax-mp 7	. 2 |- (((ps -> ch) -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))) -> (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))))
7	1, 6	ax-mp 7	1 |- (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))

Syntax hints:  -. wn 2   -> wi 3

This theorem is referenced by:  merlem8 1056

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

This theorem depends on definitions:  df-bi 163  df-an 241

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