Theorem dfnot 1261

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.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)

Assertion

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

Proof of Theorem dfnot

Step	Hyp	Ref	Expression
1		fal 1249	. 2 ⊢ ¬ ⊥
2		mtt 609	. 2 ⊢ (¬ ⊥ → (¬ φ ↔ (φ → ⊥ )))
3	1, 2	ax-mp 7	1 ⊢ (¬ φ ↔ (φ → ⊥ ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ⊥ wfal 1247

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248

This theorem is referenced by:  inegd  1262  pclem6  1264  alnex  1385  alexim  1533  difin  3168  indifdir  3187  bj-axempty2  9349