Theorem clelsb1 2292

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2165). (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 1538	. . 3 |- F/ x w e. A
2	1	sbco2 1975	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
3		nfv 1538	. . . 4 |- F/ w x e. A
4		eleq1 2250	. . . 4 |- ( w = x -> ( w e. A <-> x e. A ) )
5	3, 4	sbie 1801	. . 3 |- ( [ x / w ] w e. A <-> x e. A )
6	5	sbbii 1775	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
7		nfv 1538	. . 3 |- F/ w y e. A
8		eleq1 2250	. . 3 |- ( w = y -> ( w e. A <-> y e. A ) )
9	7, 8	sbie 1801	. 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 1772   e. wcel 2158

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183

This theorem is referenced by:  hblem  2295  nfraldya  2522  nfrexdya  2523  cbvreu  2713  sbcel1v  3037  rmo3  3066  setindel  4549  elirr  4552  en2lp  4565  zfregfr  4585  tfi  4593  bdcriota  14931