Theorem opeq2OLD

Description: Obsolete version of opeq2 as of 25-May-2024. (Contributed by NM, 25-Jun-1998) (Revised by Mario Carneiro, 26-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	opeq2OLD	|- ( 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 >. )