Theorem clelsb1f

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 ). Usage of this theorem is discouraged because it depends on ax-13 . See clelsb1fw not requiring ax-13 , but extra disjoint variables. (Contributed by Rodolfo Medina, 28-Apr-2010) (Proof shortened by Andrew Salmon, 14-Jun-2011) (Revised by Thierry Arnoux, 13-Mar-2017) (Proof shortened by Wolf Lammen, 7-May-2023) (New usage is discouraged.)

	Ref	Expression
Hypothesis	clelsb1f.1	|- F/_ x A
Assertion	clelsb1f	|- ( [ y / x ] x e. A <-> y e. A )

Proof

Step	Hyp	Ref	Expression
1		clelsb1f.1	|- F/_ x A
2	1	nfcri	|- F/ x w e. A
3	2	sbco2	|- ( [ 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 )