Theorem clelsb3 2142

Description: Substitution applied to an atomic wff (class version of elsb3 1852). (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 1421	. . 3 |- F/ y w e. A
2	1	sbco2 1839	. 2 |- ( [ x / y ] [ y / w ] w e. A <-> [ x / w ] w e. A )
3		nfv 1421	. . . 4 |- F/ w y e. A
4		eleq1 2100	. . . 4 |- ( w = y -> ( w e. A <-> y e. A ) )
5	3, 4	sbie 1674	. . 3 |- ( [ y / w ] w e. A <-> y e. A )
6	5	sbbii 1648	. 2 |- ( [ x / y ] [ y / w ] w e. A <-> [ x / y ] y e. A )
7		nfv 1421	. . 3 |- F/ w x e. A
8		eleq1 2100	. . 3 |- ( w = x -> ( w e. A <-> x e. A ) )
9	7, 8	sbie 1674	. 2 |- ( [ x / w ] w e. A <-> x e. A )
10	2, 6, 9	3bitr3i 199	1 |- ( [ x / y ] y e. A <-> x e. A )

Syntax hints:   <-> wb 98   e. wcel 1393   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036

This theorem is referenced by:  hblem  2145  nfraldya  2358  nfrexdya  2359  cbvreu  2531  sbcel1gv  2821  rmo3  2849  setindel  4263  elirr  4266  en2lp  4278  zfregfr  4298  tfi  4305  bdcriota  10003