Theorem ax2 670

Description: Standard propositional axiom derived from Lukasiewicz axioms.

Assertion

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

Proof of Theorem ax2

Step	Hyp	Ref	Expression
1		luklem7 667	. 2 |- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))
2		luklem8 668	. . 3 |- ((ps -> (ph -> ch)) -> ((ph -> ps) -> (ph -> (ph -> ch))))
3		luklem6 666	. . . 4 |- ((ph -> (ph -> ch)) -> (ph -> ch))
4		luklem8 668	. . . 4 |- (((ph -> (ph -> ch)) -> (ph -> ch)) -> (((ph -> ps) -> (ph -> (ph -> ch))) -> ((ph -> ps) -> (ph -> ch))))
5	3, 4	ax-mp 6	. . 3 |- (((ph -> ps) -> (ph -> (ph -> ch))) -> ((ph -> ps) -> (ph -> ch)))
6	2, 5	luklem1 661	. 2 |- ((ps -> (ph -> ch)) -> ((ph -> ps) -> (ph -> ch)))
7	1, 6	luklem1 661	1 |- ((ph -> (ps -> ch)) -> ((ph -> ps) -> (ph -> ch)))

Syntax hints:   -> wi 2

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

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