Theorem nannot 1529

Description: Negation in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) Use dfnan2 1524. (Revised by Wolf Lammen, 26-Jun-2020.)

Assertion

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

Proof of Theorem nannot

Step	Hyp	Ref	Expression
1		dfnan2 1524	. 2 ⊢ ((𝜑 ⊼ 𝜑) ↔ (𝜑 → ¬ 𝜑))
2		pm4.8 398	. 2 ⊢ ((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
3	1, 2	bitr2i 279	1 ⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ⊼ wnan 1521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-nan 1522

This theorem is used by:  nanbi  1530  trunantru  1611  falnanfal  1614  nic-dfneg  1703  andnand1  36953  imnand2  36954