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Metamath Proof Explorer

Theorem neleq1

Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994) (Proof shortened by Wolf Lammen, 25-Nov-2019)

Ref Expression
Assertion neleq1 ⊢ A = B → A ∉ C ↔ B ∉ C

Proof

Step Hyp Ref Expression
1 id ⊢ A = B → A = B
2 eqidd ⊢ A = B → C = C
3 1 2 neleq12d ⊢ A = B → A ∉ C ↔ B ∉ C