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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 ∈ C ↔ B ∈ D

Proof

Step Hyp Ref Expression
1 eleq1i.1 ⊢ A = B
2 eleq12i.2 ⊢ C = D
3 2 eleq2i ⊢ A ∈ C ↔ A ∈ D
4 1 eleq1i ⊢ A ∈ D ↔ B ∈ D
5 3 4 bitri ⊢ A ∈ C ↔ B ∈ D