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Metamath Proof Explorer

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 )