Theorem re1luk2 1476

Description: luk-2 1421 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1luk2	⊢ ((¬ φ → φ) → φ)

Proof of Theorem re1luk2

Step	Hyp	Ref	Expression
1		tbw-negdf 1464	. . . 4 ⊢ (((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ )
2		tbw-ax2 1466	. . . . 5 ⊢ ((((φ → ⊥ ) → ¬ φ) → ⊥ ) → ((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )))
3		tbwlem4 1473	. . . . 5 ⊢ (((((φ → ⊥ ) → ¬ φ) → ⊥ ) → ((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ ))) → ((((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ ) → ((φ → ⊥ ) → ¬ φ)))
4	2, 3	ax-mp 5	. . . 4 ⊢ ((((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ ) → ((φ → ⊥ ) → ¬ φ))
5	1, 4	ax-mp 5	. . 3 ⊢ ((φ → ⊥ ) → ¬ φ)
6		tbw-ax1 1465	. . 3 ⊢ (((φ → ⊥ ) → ¬ φ) → ((¬ φ → φ) → ((φ → ⊥ ) → φ)))
7	5, 6	ax-mp 5	. 2 ⊢ ((¬ φ → φ) → ((φ → ⊥ ) → φ))
8		tbw-ax3 1467	. 2 ⊢ (((φ → ⊥ ) → φ) → φ)
9	7, 8	tbwsyl 1469	1 ⊢ ((¬ φ → φ) → φ)

Syntax hints:  ¬ wn 3   → wi 4   ⊥ wfal 1317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320

This theorem is referenced by: (None)