Theorem negcon1

Description: Negative contraposition law. (Contributed by NM, 9-May-2004)

	Ref	Expression
Assertion	negcon1	|- ( ( A e. CC /\ B e. CC ) -> ( -u A = B <-> -u B = A ) )

Proof

Step	Hyp	Ref	Expression
1		negcl	|- ( A e. CC -> -u A e. CC )
2		neg11	|- ( ( -u A e. CC /\ B e. CC ) -> ( -u -u A = -u B <-> -u A = B ) )
3	1 2	sylan	|- ( ( A e. CC /\ B e. CC ) -> ( -u -u A = -u B <-> -u A = B ) )
4		negneg	|- ( A e. CC -> -u -u A = A )
5	4	adantr	|- ( ( A e. CC /\ B e. CC ) -> -u -u A = A )
6	5	eqeq1d	|- ( ( A e. CC /\ B e. CC ) -> ( -u -u A = -u B <-> A = -u B ) )
7	3 6	bitr3d	|- ( ( A e. CC /\ B e. CC ) -> ( -u A = B <-> A = -u B ) )
8		eqcom	|- ( A = -u B <-> -u B = A )
9	7 8	bitrdi	|- ( ( A e. CC /\ B e. CC ) -> ( -u A = B <-> -u B = A ) )