Theorem altopth2

Description: Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012)

	Ref	Expression
Assertion	altopth2	|- ( B e. V -> ( << A , B >> = << C , D >> -> B = D ) )

Proof

Step	Hyp	Ref	Expression
1		altopthsn	|- ( << A , B >> = << C , D >> <-> ( { A } = { C } /\ { B } = { D } ) )
2		sneqrg	|- ( B e. V -> ( { B } = { D } -> B = D ) )
3	2	adantld	|- ( B e. V -> ( ( { A } = { C } /\ { B } = { D } ) -> B = D ) )
4	1 3	biimtrid	|- ( B e. V -> ( << A , B >> = << C , D >> -> B = D ) )