Theorem tbw-negdf 1464

Description: The definition of negation, in terms of → and ⊥. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tbw-negdf	⊢ (((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ )

Proof of Theorem tbw-negdf

Step	Hyp	Ref	Expression
1		pm2.21 100	. . 3 ⊢ (¬ φ → (φ → ⊥ ))
2		ax-1 6	. . . . 5 ⊢ (¬ φ → ((φ → ⊥ ) → ¬ φ))
3		falim 1328	. . . . 5 ⊢ ( ⊥ → ((φ → ⊥ ) → ¬ φ))
4	2, 3	ja 153	. . . 4 ⊢ ((φ → ⊥ ) → ((φ → ⊥ ) → ¬ φ))
5	4	pm2.43i 43	. . 3 ⊢ ((φ → ⊥ ) → ¬ φ)
6	1, 5	impbii 180	. 2 ⊢ (¬ φ ↔ (φ → ⊥ ))
7		tbw-bijust 1463	. 2 ⊢ ((¬ φ ↔ (φ → ⊥ )) ↔ (((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ ))
8	6, 7	mpbi 199	1 ⊢ (((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ⊥ 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:  re1luk2  1476  re1luk3  1477