Theorem dfnot 1415

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 1404	. 2 ⊢ ¬ ⊥
2		mtt 691	. 2 ⊢ (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥)))
3	1, 2	ax-mp 5	1 ⊢ (¬ 𝜑 ↔ (𝜑 → ⊥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ⊥wfal 1402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403

This theorem is referenced by:  inegd  1416  pclem6  1418  dcfrompeirce  1494  alnex  1547  alexim  1693  difin  3444  indifdir  3463  recvguniq  11555  logbgcd1irr  15690  bj-axempty2  16489  pw1ndom3  16589