Theorem ax2 1432

Description: Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax2	|- ((ph -> (ps -> ch)) -> ((ph -> ps) -> (ph -> ch)))

Proof of Theorem ax2

Step	Hyp	Ref	Expression
1		luklem7 1429	. 2 |- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))
2		luklem8 1430	. . 3 |- ((ps -> (ph -> ch)) -> ((ph -> ps) -> (ph -> (ph -> ch))))
3		luklem6 1428	. . . 4 |- ((ph -> (ph -> ch)) -> (ph -> ch))
4		luklem8 1430	. . . 4 |- (((ph -> (ph -> ch)) -> (ph -> ch)) -> (((ph -> ps) -> (ph -> (ph -> ch))) -> ((ph -> ps) -> (ph -> ch))))
5	3, 4	ax-mp 5	. . 3 |- (((ph -> ps) -> (ph -> (ph -> ch))) -> ((ph -> ps) -> (ph -> ch)))
6	2, 5	luklem1 1423	. 2 |- ((ps -> (ph -> ch)) -> ((ph -> ps) -> (ph -> ch)))
7	1, 6	luklem1 1423	1 |- ((ph -> (ps -> ch)) -> ((ph -> ps) -> (ph -> ch)))

Syntax hints:   -> wi 4

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

This theorem is referenced by: (None)