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Metamath Proof Explorer

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) Avoid ax-13 . (Revised by GG, 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 )