Theorem df3nandALT2 31051

Description: The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)

Assertion

Ref	Expression
df3nandALT2	|- ( ( ph -/\ ps -/\ ch ) <-> -. ( ph /\ ps /\ ch ) )

Proof of Theorem df3nandALT2

Step	Hyp	Ref	Expression
1		df-3nand 31049	. 2 |- ( ( ph -/\ ps -/\ ch ) <-> ( ph -> ( ps -> -. ch ) ) )
2		imnan 423	. . 3 |- ( ( ps -> -. ch ) <-> -. ( ps /\ ch ) )
3	2	imbi2i 313	. 2 |- ( ( ph -> ( ps -> -. ch ) ) <-> ( ph -> -. ( ps /\ ch ) ) )
4		imnan 423	. . 3 |- ( ( ph -> -. ( ps /\ ch ) ) <-> -. ( ph /\ ( ps /\ ch ) ) )
5		3anass 986	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
6	4, 5	xchbinxr 312	. 2 |- ( ( ph -> -. ( ps /\ ch ) ) <-> -. ( ph /\ ps /\ ch ) )
7	1, 3, 6	3bitri 274	1 |- ( ( ph -/\ ps -/\ ch ) <-> -. ( ph /\ ps /\ ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   -/\ w3nand 31048

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-3nand 31049

This theorem is referenced by: (None)