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	⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))

Proof of Theorem ax2

Step	Hyp	Ref	Expression
1		luklem7 1429	. 2 ⊢ ((φ → (ψ → χ)) → (ψ → (φ → χ)))
2		luklem8 1430	. . 3 ⊢ ((ψ → (φ → χ)) → ((φ → ψ) → (φ → (φ → χ))))
3		luklem6 1428	. . . 4 ⊢ ((φ → (φ → χ)) → (φ → χ))
4		luklem8 1430	. . . 4 ⊢ (((φ → (φ → χ)) → (φ → χ)) → (((φ → ψ) → (φ → (φ → χ))) → ((φ → ψ) → (φ → χ))))
5	3, 4	ax-mp 5	. . 3 ⊢ (((φ → ψ) → (φ → (φ → χ))) → ((φ → ψ) → (φ → χ)))
6	2, 5	luklem1 1423	. 2 ⊢ ((ψ → (φ → χ)) → ((φ → ψ) → (φ → χ)))
7	1, 6	luklem1 1423	1 ⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-meredith 1406

This theorem is used by: (None)