Theorem hadnot 1501

Description: The adder sum distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Assertion

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

Proof of Theorem hadnot

Step	Hyp	Ref	Expression
1		notbi 297	. . 3 |- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )
2	1	bibi1i 316	. 2 |- ( ( ( ph <-> ps ) <-> -. ch ) <-> ( ( -. ph <-> -. ps ) <-> -. ch ) )
3		xor3 359	. . 3 |- ( -. ( ( ph <-> ps ) <-> ch ) <-> ( ( ph <-> ps ) <-> -. ch ) )
4		hadbi 1497	. . 3 |- (hadd ( ph , ps , ch ) <-> ( ( ph <-> ps ) <-> ch ) )
5	3, 4	xchnxbir 311	. 2 |- ( -. hadd ( ph , ps , ch ) <-> ( ( ph <-> ps ) <-> -. ch ) )
6		hadbi 1497	. 2 |- (hadd ( -. ph , -. ps , -. ch ) <-> ( ( -. ph <-> -. ps ) <-> -. ch ) )
7	2, 5, 6	3bitr4i 281	1 |- ( -. hadd ( ph , ps , ch ) <-> hadd ( -. ph , -. ps , -. ch ) )

Syntax hints:   -. wn 3   <-> wb 188  haddwhad 1492

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 189  df-xor 1402  df-had 1493

This theorem is referenced by:  had0  1503