Theorem elsb2

Description: Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 for substitution for the first argument. (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	elsb2	|- ( [ 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 )