Theorem nannot 1445

Description: Show equivalence between negation and the Nicod version. To derive nic-dfneg 1586, apply nanbi 1446. (Contributed by Jeff Hoffman, 19-Nov-2007.)

Assertion

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

Proof of Theorem nannot

Step	Hyp	Ref	Expression
1		df-nan 1440	. . 3 ⊢ ((𝜓 ⊼ 𝜓) ↔ ¬ (𝜓 ∧ 𝜓))
2		anidm 674	. . 3 ⊢ ((𝜓 ∧ 𝜓) ↔ 𝜓)
3	1, 2	xchbinx 323	. 2 ⊢ ((𝜓 ⊼ 𝜓) ↔ ¬ 𝜓)
4	3	bicomi 213	1 ⊢ (¬ 𝜓 ↔ (𝜓 ⊼ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   ⊼ wnan 1439

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 196  df-an 385  df-nan 1440

This theorem is referenced by:  nanbi  1446  trunantru  1515  falnanfal  1518  nic-dfneg  1586  andnand1  31568  imnand2  31569