Theorem ax2 670

Description: Standard propositional axiom derived from Lukasiewicz axioms.

Assertion

Ref	Expression
ax2	⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))

Proof of Theorem ax2

Step	Hyp	Ref	Expression
1		luklem7 667	. 2 ⊢ ((φ → (ψ → χ)) → (ψ → (φ → χ)))
2		luklem8 668	. . 3 ⊢ ((ψ → (φ → χ)) → ((φ → ψ) → (φ → (φ → χ))))
3		luklem6 666	. . . 4 ⊢ ((φ → (φ → χ)) → (φ → χ))
4		luklem8 668	. . . 4 ⊢ (((φ → (φ → χ)) → (φ → χ)) → (((φ → ψ) → (φ → (φ → χ))) → ((φ → ψ) → (φ → χ))))
5	3, 4	ax-mp 6	. . 3 ⊢ (((φ → ψ) → (φ → (φ → χ))) → ((φ → ψ) → (φ → χ)))
6	2, 5	luklem1 661	. 2 ⊢ ((ψ → (φ → χ)) → ((φ → ψ) → (φ → χ)))
7	1, 6	luklem1 661	1 ⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))

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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