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Metamath Proof Explorer

Theorem eleq12

Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999)

Ref Expression
Assertion eleq12 ⊢ A = B ∧ C = D → A ∈ C ↔ B ∈ D

Proof

Step Hyp Ref Expression
1 eleq1 ⊢ A = B → A ∈ C ↔ B ∈ C
2 eleq2 ⊢ C = D → B ∈ C ↔ B ∈ D
3 1 2 sylan9bb ⊢ A = B ∧ C = D → A ∈ C ↔ B ∈ D