Theorem hadnot 1436

Description: The half adder distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
hadnot	|- ( -. hadd ( ph , ps , ch ) <-> hadd ( -. ph , -. ps , -. ch ) )

Proof of Theorem hadnot

Step	Hyp	Ref	Expression
1		xorneg 1363	. . . 4 |- ( ( -. ph \/_ -. ps ) <-> ( ph \/_ ps ) )
2		biid 236	. . . 4 |- ( -. ch <-> -. ch )
3	1, 2	xorbi12i 1364	. . 3 |- ( ( ( -. ph \/_ -. ps ) \/_ -. ch ) <-> ( ( ph \/_ ps ) \/_ -. ch ) )
4		xorneg2 1362	. . 3 |- ( ( ( ph \/_ ps ) \/_ -. ch ) <-> -. ( ( ph \/_ ps ) \/_ ch ) )
5	3, 4	bitr2i 250	. 2 |- ( -. ( ( ph \/_ ps ) \/_ ch ) <-> ( ( -. ph \/_ -. ps ) \/_ -. ch ) )
6		df-had 1422	. . 3 |- (hadd ( ph , ps , ch ) <-> ( ( ph \/_ ps ) \/_ ch ) )
7	6	notbii 296	. 2 |- ( -. hadd ( ph , ps , ch ) <-> -. ( ( ph \/_ ps ) \/_ ch ) )
8		df-had 1422	. 2 |- (hadd ( -. ph , -. ps , -. ch ) <-> ( ( -. ph \/_ -. ps ) \/_ -. ch ) )
9	5, 7, 8	3bitr4i 277	1 |- ( -. hadd ( ph , ps , ch ) <-> hadd ( -. ph , -. ps , -. ch ) )

Syntax hints:   -. wn 3   <-> wb 184   \/_ wxo 1351  haddwhad 1420

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 185  df-xor 1352  df-had 1422

This theorem is referenced by:  had0  1446