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Metamath Proof Explorer

Theorem eleq2

Description: Equality implies equivalence of membership. (Contributed by NM, 26-May-1993) (Proof shortened by Wolf Lammen, 20-Nov-2019)

Ref Expression
Assertion eleq2 ⊢ A = B → C ∈ A ↔ C ∈ B

Proof

Step Hyp Ref Expression
1 id ⊢ A = B → A = B
2 1 eleq2d ⊢ A = B → C ∈ A ↔ C ∈ B