Theorem clelsb1 2312

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2185). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
clelsb1	|- ( [ y / x ] x e. A <-> y e. A )

Distinct variable group:   x, A

Allowed substitution hint:   A( y)

Proof of Theorem clelsb1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1552	. . 3 |- F/ x w e. A
2	1	sbco2 1994	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
3		nfv 1552	. . . 4 |- F/ w x e. A
4		eleq1 2270	. . . 4 |- ( w = x -> ( w e. A <-> x e. A ) )
5	3, 4	sbie 1815	. . 3 |- ( [ x / w ] w e. A <-> x e. A )
6	5	sbbii 1789	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
7		nfv 1552	. . 3 |- F/ w y e. A
8		eleq1 2270	. . 3 |- ( w = y -> ( w e. A <-> y e. A ) )
9	7, 8	sbie 1815	. 2 |- ( [ y / w ] w e. A <-> y e. A )
10	2, 6, 9	3bitr3i 210	1 |- ( [ y / x ] x e. A <-> y e. A )

Syntax hints:   <-> wb 105   [wsb 1786   e. wcel 2178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203

This theorem is referenced by:  hblem  2315  eqabdv  2336  nfraldya  2543  nfrexdya  2544  cbvreu  2740  sbcel1v  3068  rmo3  3098  setindel  4604  elirr  4607  en2lp  4620  zfregfr  4640  tfi  4648  bdcriota  16018