# Metamath Proof Explorer

## 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 >. )`