Theorem luklem4 1426

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
luklem4	⊢ ((((¬ φ → φ) → φ) → ψ) → ψ)

Proof of Theorem luklem4

Step	Hyp	Ref	Expression
1		luk-2 1421	. . . 4 ⊢ ((¬ ((¬ φ → φ) → φ) → ((¬ φ → φ) → φ)) → ((¬ φ → φ) → φ))
2		luk-2 1421	. . . . 5 ⊢ ((¬ φ → φ) → φ)
3		luklem3 1425	. . . . 5 ⊢ (((¬ φ → φ) → φ) → (((¬ ((¬ φ → φ) → φ) → ((¬ φ → φ) → φ)) → ((¬ φ → φ) → φ)) → (¬ ψ → ((¬ φ → φ) → φ))))
4	2, 3	ax-mp 5	. . . 4 ⊢ (((¬ ((¬ φ → φ) → φ) → ((¬ φ → φ) → φ)) → ((¬ φ → φ) → φ)) → (¬ ψ → ((¬ φ → φ) → φ)))
5	1, 4	ax-mp 5	. . 3 ⊢ (¬ ψ → ((¬ φ → φ) → φ))
6		luk-1 1420	. . 3 ⊢ ((¬ ψ → ((¬ φ → φ) → φ)) → ((((¬ φ → φ) → φ) → ψ) → (¬ ψ → ψ)))
7	5, 6	ax-mp 5	. 2 ⊢ ((((¬ φ → φ) → φ) → ψ) → (¬ ψ → ψ))
8		luk-2 1421	. 2 ⊢ ((¬ ψ → ψ) → ψ)
9	7, 8	luklem1 1423	1 ⊢ ((((¬ φ → φ) → φ) → ψ) → ψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

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

This theorem is used by:  luklem5  1427  luklem6  1428  ax3  1433