Metamath Proof Explorer

Theorem elsb4

Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010) (Proof shortened by Andrew Salmon, 14-Jun-2011) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023)

Ref Expression
Assertion elsb4 
|- ( [ y / x ] z e. x <-> z e. y )

Proof

Step Hyp Ref Expression
1 elequ2 
 |-  ( x = w -> ( z e. x <-> z e. w ) )
2 elequ2 
 |-  ( w = y -> ( z e. w <-> z e. y ) )
3 1 2 sbievw2 
 |-  ( [ y / x ] z e. x <-> z e. y )