Theorem clelsb3 2192

Description: Substitution applied to an atomic wff (class version of elsb3 1900). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
clelsb3	|- ( [ x / y ] y e. A <-> x e. A )

Distinct variable group:   y, A

Allowed substitution hint:   A( x)

Proof of Theorem clelsb3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1466	. . 3 |- F/ y w e. A
2	1	sbco2 1887	. 2 |- ( [ x / y ] [ y / w ] w e. A <-> [ x / w ] w e. A )
3		nfv 1466	. . . 4 |- F/ w y e. A
4		eleq1 2150	. . . 4 |- ( w = y -> ( w e. A <-> y e. A ) )
5	3, 4	sbie 1721	. . 3 |- ( [ y / w ] w e. A <-> y e. A )
6	5	sbbii 1695	. 2 |- ( [ x / y ] [ y / w ] w e. A <-> [ x / y ] y e. A )
7		nfv 1466	. . 3 |- F/ w x e. A
8		eleq1 2150	. . 3 |- ( w = x -> ( w e. A <-> x e. A ) )
9	7, 8	sbie 1721	. 2 |- ( [ x / w ] w e. A <-> x e. A )
10	2, 6, 9	3bitr3i 208	1 |- ( [ x / y ] y e. A <-> x e. A )

Syntax hints:   <-> wb 103   e. wcel 1438   [wsb 1692

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084

This theorem is referenced by:  hblem  2195  nfraldya  2412  nfrexdya  2413  cbvreu  2588  sbcel1v  2901  rmo3  2930  setindel  4354  elirr  4357  en2lp  4370  zfregfr  4389  tfi  4397  bdcriota  11774