Theorem dfnot 1314

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 1303	. 2 ⊢ ¬ ⊥
2		mtt 648	. 2 ⊢ (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥)))
3	1, 2	ax-mp 7	1 ⊢ (¬ 𝜑 ↔ (𝜑 → ⊥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ⊥wfal 1301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302

This theorem is referenced by:  inegd  1315  pclem6  1317  alnex  1440  alexim  1588  difin  3252  indifdir  3271  recvguniq  10543  bj-axempty2  12502