Theorem andnand1 32398

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

Assertion

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

Proof of Theorem andnand1

Step	Hyp	Ref	Expression
1		3anass 1042	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
2		pm4.63 437	. . . 4 |- ( -. ( ps -> -. ch ) <-> ( ps /\ ch ) )
3	2	anbi2i 730	. . 3 |- ( ( ph /\ -. ( ps -> -. ch ) ) <-> ( ph /\ ( ps /\ ch ) ) )
4		annim 441	. . 3 |- ( ( ph /\ -. ( ps -> -. ch ) ) <-> -. ( ph -> ( ps -> -. ch ) ) )
5	1, 3, 4	3bitr2i 288	. 2 |- ( ( ph /\ ps /\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )
6		df-3nand 32395	. . 3 |- ( ( ph -/\ ps -/\ ch ) <-> ( ph -> ( ps -> -. ch ) ) )
7	6	notbii 310	. 2 |- ( -. ( ph -/\ ps -/\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )
8		nannot 1453	. 2 |- ( -. ( ph -/\ ps -/\ ch ) <-> ( ( ph -/\ ps -/\ ch ) -/\ ( ph -/\ ps -/\ ch ) ) )
9	5, 7, 8	3bitr2i 288	1 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph -/\ ps -/\ ch ) -/\ ( ph -/\ ps -/\ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   -/\ wnan 1447   -/\ w3nand 32394

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 197  df-an 386  df-3an 1039  df-nan 1448  df-3nand 32395

This theorem is referenced by: (None)