Theorem dfnot 1262

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 1250	. 2 ⊢ ¬ ⊥
2		mtt 610	. 2 ⊢ (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥)))
3	1, 2	ax-mp 7	1 ⊢ (¬ 𝜑 ↔ (𝜑 → ⊥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ⊥wfal 1248

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 1246  df-fal 1249

This theorem is referenced by:  inegd  1263  pclem6  1265  alnex  1388  alexim  1536  difin  3171  indifdir  3190  recvguniq  9433  bj-axempty2  9849