Theorem clelsb1fw

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 ). Version of clelsb1f with a disjoint variable condition, which does not require ax-13 . (Contributed by Rodolfo Medina, 28-Apr-2010) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypothesis	clelsb1fw.1	|- F/_ x A
Assertion	clelsb1fw	|- ( [ y / x ] x e. A <-> y e. A )

Proof

Step	Hyp	Ref	Expression
1		clelsb1fw.1	|- F/_ x A
2	1	nfcri	|- F/ x w e. A
3	2	sbco2v	|- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
4		clelsb1	|- ( [ x / w ] w e. A <-> x e. A )
5	4	sbbii	|- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
6		clelsb1	|- ( [ y / w ] w e. A <-> y e. A )
7	3 5 6	3bitr3i	|- ( [ y / x ] x e. A <-> y e. A )