Theorem andnand1 31064

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 986	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
2		pm4.63 422	. . . 4 |- ( -. ( ps -> -. ch ) <-> ( ps /\ ch ) )
3	2	anbi2i 698	. . 3 |- ( ( ph /\ -. ( ps -> -. ch ) ) <-> ( ph /\ ( ps /\ ch ) ) )
4		annim 426	. . 3 |- ( ( ph /\ -. ( ps -> -. ch ) ) <-> -. ( ph -> ( ps -> -. ch ) ) )
5	1, 3, 4	3bitr2i 276	. 2 |- ( ( ph /\ ps /\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )
6		df-3nand 31061	. . 3 |- ( ( ph -/\ ps -/\ ch ) <-> ( ph -> ( ps -> -. ch ) ) )
7	6	notbii 297	. 2 |- ( -. ( ph -/\ ps -/\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )
8		nannot 1387	. 2 |- ( -. ( ph -/\ ps -/\ ch ) <-> ( ( ph -/\ ps -/\ ch ) -/\ ( ph -/\ ps -/\ ch ) ) )
9	5, 7, 8	3bitr2i 276	1 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph -/\ ps -/\ ch ) -/\ ( ph -/\ ps -/\ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   -/\ wnan 1379   -/\ w3nand 31060

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 188  df-an 372  df-3an 984  df-nan 1380  df-3nand 31061

This theorem is referenced by: (None)