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 e. A <-> y e. A )

Proof

Step Hyp Ref Expression
1 eleq1w 
 |-  ( x = w -> ( x e. A <-> w e. A ) )
2 eleq1w 
 |-  ( w = y -> ( w e. A <-> y e. A ) )
3 1 2 sbievw2 
 |-  ( [ y / x ] x e. A <-> y e. A )