Theorem dfnot 1332

Description: Given falsum, we can define the negation of a wff φ as the statement that a contradiction follows from assuming φ. (Contributed by Mario Carneiro, 9-Feb-2017.)

Assertion

Ref	Expression
dfnot	⊢ (¬ φ ↔ (φ → ⊥ ))

Proof of Theorem dfnot

Step	Hyp	Ref	Expression
1		pm2.21 100	. 2 ⊢ (¬ φ → (φ → ⊥ ))
2		id 19	. . 3 ⊢ (¬ φ → ¬ φ)
3		falim 1328	. . 3 ⊢ ( ⊥ → ¬ φ)
4	2, 3	ja 153	. 2 ⊢ ((φ → ⊥ ) → ¬ φ)
5	1, 4	impbii 180	1 ⊢ (¬ 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:  inegd  1333