Theorem xorneg1 1413

Description: \/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)

Assertion

Ref	Expression
xorneg1	|- ( ( -. ph \/_ ps ) <-> -. ( ph \/_ ps ) )

Proof of Theorem xorneg1

Step	Hyp	Ref	Expression
1		xorcom 1404	. 2 |- ( ( -. ph \/_ ps ) <-> ( ps \/_ -. ph ) )
2		xorneg2 1412	. . 3 |- ( ( ps \/_ -. ph ) <-> -. ( ps \/_ ph ) )
3		xorcom 1404	. . 3 |- ( ( ps \/_ ph ) <-> ( ph \/_ ps ) )
4	2, 3	xchbinx 312	. 2 |- ( ( ps \/_ -. ph ) <-> -. ( ph \/_ ps ) )
5	1, 4	bitri 253	1 |- ( ( -. ph \/_ ps ) <-> -. ( ph \/_ ps ) )

Syntax hints:   -. wn 3   <-> wb 188   \/_ wxo 1401

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

This theorem is referenced by:  xorneg2OLD  1415  xorneg  1416