Theorem andnand1

Description: Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011)

	Ref	Expression
Assertion	andnand1	|- ( ( ph /\ ps /\ ch ) <-> ( ( ph -/\ ps -/\ ch ) -/\ ( ph -/\ ps -/\ ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		3anass	|- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
2		pm4.63	|- ( -. ( ps -> -. ch ) <-> ( ps /\ ch ) )
3	2	anbi2i	|- ( ( ph /\ -. ( ps -> -. ch ) ) <-> ( ph /\ ( ps /\ ch ) ) )
4		annim	|- ( ( ph /\ -. ( ps -> -. ch ) ) <-> -. ( ph -> ( ps -> -. ch ) ) )
5	1 3 4	3bitr2i	|- ( ( ph /\ ps /\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )
6		df-3nand	|- ( ( ph -/\ ps -/\ ch ) <-> ( ph -> ( ps -> -. ch ) ) )
7	6	notbii	|- ( -. ( ph -/\ ps -/\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )
8		nannot	|- ( -. ( ph -/\ ps -/\ ch ) <-> ( ( ph -/\ ps -/\ ch ) -/\ ( ph -/\ ps -/\ ch ) ) )
9	5 7 8	3bitr2i	|- ( ( ph /\ ps /\ ch ) <-> ( ( ph -/\ ps -/\ ch ) -/\ ( ph -/\ ps -/\ ch ) ) )