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Metamath Proof Explorer

Theorem opeq2

Description: Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998) (Revised by Mario Carneiro, 26-Apr-2015)

Ref Expression
Assertion opeq2 
|- ( A = B -> <. C , A >. = <. C , B >. )

Proof

Step Hyp Ref Expression
1 eleq1 
 |-  ( A = B -> ( A e. _V <-> B e. _V ) )
2 1 anbi2d 
 |-  ( A = B -> ( ( C e. _V /\ A e. _V ) <-> ( C e. _V /\ B e. _V ) ) )
3 preq2 
 |-  ( A = B -> { C , A } = { C , B } )
4 3 preq2d 
 |-  ( A = B -> { { C } , { C , A } } = { { C } , { C , B } } )
5 2 4 ifbieq1d 
 |-  ( A = B -> if ( ( C e. _V /\ A e. _V ) , { { C } , { C , A } } , (/) ) = if ( ( C e. _V /\ B e. _V ) , { { C } , { C , B } } , (/) ) )
6 dfopif 
 |-  <. C , A >. = if ( ( C e. _V /\ A e. _V ) , { { C } , { C , A } } , (/) )
7 dfopif 
 |-  <. C , B >. = if ( ( C e. _V /\ B e. _V ) , { { C } , { C , B } } , (/) )
8 5 6 7 3eqtr4g 
 |-  ( A = B -> <. C , A >. = <. C , B >. )