Theorem clelsb1 2343

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2216). (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 1581	. . 3 |- F/ x w e. A
2	1	sbco2 2025	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
3		nfv 1581	. . . 4 |- F/ w x e. A
4		eleq1 2301	. . . 4 |- ( w = x -> ( w e. A <-> x e. A ) )
5	3, 4	sbie 1844	. . 3 |- ( [ x / w ] w e. A <-> x e. A )
6	5	sbbii 1818	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
7		nfv 1581	. . 3 |- F/ w y e. A
8		eleq1 2301	. . 3 |- ( w = y -> ( w e. A <-> y e. A ) )
9	7, 8	sbie 1844	. 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 1815   e. wcel 2209

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234

This theorem is referenced by:  hblem  2346  eqabdv  2369  nfraldya  2585  nfrexdya  2586  cbvreu  2784  sbcel1v  3114  rmo3  3144  setindel  4680  elirr  4683  en2lp  4696  zfregfr  4716  tfi  4724  bdcriota  16823