Theorem clelsb1 2339

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2212). (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 1577	. . 3 |- F/ x w e. A
2	1	sbco2 2021	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
3		nfv 1577	. . . 4 |- F/ w x e. A
4		eleq1 2297	. . . 4 |- ( w = x -> ( w e. A <-> x e. A ) )
5	3, 4	sbie 1840	. . 3 |- ( [ x / w ] w e. A <-> x e. A )
6	5	sbbii 1814	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
7		nfv 1577	. . 3 |- F/ w y e. A
8		eleq1 2297	. . 3 |- ( w = y -> ( w e. A <-> y e. A ) )
9	7, 8	sbie 1840	. 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 1811   e. wcel 2205

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-cleq 2227  df-clel 2230

This theorem is referenced by:  hblem  2342  eqabdv  2365  nfraldya  2579  nfrexdya  2580  cbvreu  2778  sbcel1v  3108  rmo3  3138  setindel  4665  elirr  4668  en2lp  4681  zfregfr  4701  tfi  4709  bdcriota  16779