Theorem clelsb1 2282

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2155). (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 1528	. . 3 |- F/ x w e. A
2	1	sbco2 1965	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
3		nfv 1528	. . . 4 |- F/ w x e. A
4		eleq1 2240	. . . 4 |- ( w = x -> ( w e. A <-> x e. A ) )
5	3, 4	sbie 1791	. . 3 |- ( [ x / w ] w e. A <-> x e. A )
6	5	sbbii 1765	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
7		nfv 1528	. . 3 |- F/ w y e. A
8		eleq1 2240	. . 3 |- ( w = y -> ( w e. A <-> y e. A ) )
9	7, 8	sbie 1791	. 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 1762   e. wcel 2148

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173

This theorem is referenced by:  hblem  2285  nfraldya  2512  nfrexdya  2513  cbvreu  2702  sbcel1v  3026  rmo3  3055  setindel  4538  elirr  4541  en2lp  4554  zfregfr  4574  tfi  4582  bdcriota  14638