Theorem altopthg

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012)

	Ref	Expression
Assertion	altopthg	|- ( ( A e. V /\ B e. W ) -> ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) ) )

Proof

Step	Hyp	Ref	Expression
1		altopthsn	|- ( << A , B >> = << C , D >> <-> ( { A } = { C } /\ { B } = { D } ) )
2		sneqbg	|- ( A e. V -> ( { A } = { C } <-> A = C ) )
3		sneqbg	|- ( B e. W -> ( { B } = { D } <-> B = D ) )
4	2 3	bi2anan9	|- ( ( A e. V /\ B e. W ) -> ( ( { A } = { C } /\ { B } = { D } ) <-> ( A = C /\ B = D ) ) )
5	1 4	syl5bb	|- ( ( A e. V /\ B e. W ) -> ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) ) )