Theorem nannan 1291

Description: Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.)

Assertion

Ref	Expression
nannan	⊢ ((φ ⊼ (χ ⊼ ψ)) ↔ (φ → (χ ∧ ψ)))

Proof of Theorem nannan

Step	Hyp	Ref	Expression
1		df-nan 1288	. . 3 ⊢ ((φ ⊼ (χ ⊼ ψ)) ↔ ¬ (φ ∧ (χ ⊼ ψ)))
2		df-nan 1288	. . . 4 ⊢ ((χ ⊼ ψ) ↔ ¬ (χ ∧ ψ))
3	2	anbi2i 675	. . 3 ⊢ ((φ ∧ (χ ⊼ ψ)) ↔ (φ ∧ ¬ (χ ∧ ψ)))
4	1, 3	xchbinx 301	. 2 ⊢ ((φ ⊼ (χ ⊼ ψ)) ↔ ¬ (φ ∧ ¬ (χ ∧ ψ)))
5		iman 413	. 2 ⊢ ((φ → (χ ∧ ψ)) ↔ ¬ (φ ∧ ¬ (χ ∧ ψ)))
6	4, 5	bitr4i 243	1 ⊢ ((φ ⊼ (χ ⊼ ψ)) ↔ (φ → (χ ∧ ψ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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:  nanim  1292  nic-mp  1436  nic-ax  1438