Theorem nannot 1499

Description: Negation in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) Use dfnan2 1494. (Revised by Wolf Lammen, 26-Jun-2020.)

Assertion

Ref	Expression
nannot	⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑))

Proof of Theorem nannot

Step	Hyp	Ref	Expression
1		dfnan2 1494	. 2 ⊢ ((𝜑 ⊼ 𝜑) ↔ (𝜑 → ¬ 𝜑))
2		pm4.8 392	. 2 ⊢ ((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
3	1, 2	bitr2i 276	1 ⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ⊼ wnan 1491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-nan 1492

This theorem is referenced by:  nanbi  1500  trunantru  1581  falnanfal  1584  nic-dfneg  1670  andnand1  36402  imnand2  36403