Theorem nannot 1293

Description: Show equivalence between negation and the Nicod version. To derive nic-dfneg 1435, apply nanbi 1294. (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 1288	. . 3 ⊢ ((ψ ⊼ ψ) ↔ ¬ (ψ ∧ ψ))
2		anidm 625	. . 3 ⊢ ((ψ ∧ ψ) ↔ ψ)
3	1, 2	xchbinx 301	. 2 ⊢ ((ψ ⊼ ψ) ↔ ¬ ψ)
4	3	bicomi 193	1 ⊢ (¬ ψ ↔ (ψ ⊼ ψ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287

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 177  df-an 360  df-nan 1288

This theorem is referenced by:  nanbi  1294  trunantru  1354  falnanfal  1357  nic-dfneg  1435