Theorem dfnot 1303

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 1292	. 2 ⊢ ¬ ⊥
2		mtt 643	. 2 ⊢ (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥)))
3	1, 2	ax-mp 7	1 ⊢ (¬ 𝜑 ↔ (𝜑 → ⊥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  ⊥wfal 1290

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291

This theorem is referenced by:  inegd  1304  pclem6  1306  alnex  1429  alexim  1577  difin  3218  indifdir  3237  recvguniq  10107  bj-axempty2  10977