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Metamath Proof Explorer

Theorem eleq12i

Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994)

Ref Expression
Hypotheses eleq1i.1 
|- A = B
eleq12i.2 
|- C = D
Assertion eleq12i 
|- ( A e. C <-> B e. D )

Proof

Step Hyp Ref Expression
1 eleq1i.1 
 |-  A = B
2 eleq12i.2 
 |-  C = D
3 2 eleq2i 
 |-  ( A e. C <-> A e. D )
4 1 eleq1i 
 |-  ( A e. D <-> B e. D )
5 3 4 bitri 
 |-  ( A e. C <-> B e. D )