Metamath Proof Explorer


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 syl5bi
 |-  ( B e. V -> ( << A , B >> = << C , D >> -> B = D ) )