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Metamath Proof Explorer

Theorem clelsb1

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 ). (Contributed by Rodolfo Medina, 28-Apr-2010) (Proof shortened by Andrew Salmon, 14-Jun-2011)

Ref Expression
Assertion clelsb1 ⊢ y x x ∈ A ↔ y ∈ A

Proof

Step Hyp Ref Expression
1 eleq1w ⊢ x = w → x ∈ A ↔ w ∈ A
2 eleq1w ⊢ w = y → w ∈ A ↔ y ∈ A
3 1 2 sbievw2 ⊢ y x x ∈ A ↔ y ∈ A