Theorem clelsb1f 2311

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2143). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)

Hypothesis

Ref	Expression
clelsb1f.1	|- F/_ x A

Assertion

Ref	Expression
clelsb1f	|- ( [ y / x ] x e. A <-> y e. A )

Proof of Theorem clelsb1f

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb1f.1	. . . 4 |- F/_ x A
2	1	nfcri 2301	. . 3 |- F/ x w e. A
3	2	sbco2 1953	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
4		nfv 1516	. . . 4 |- F/ w x e. A
5		eleq1w 2226	. . . 4 |- ( w = x -> ( w e. A <-> x e. A ) )
6	4, 5	sbie 1779	. . 3 |- ( [ x / w ] w e. A <-> x e. A )
7	6	sbbii 1753	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
8		nfv 1516	. . 3 |- F/ w y e. A
9		eleq1w 2226	. . 3 |- ( w = y -> ( w e. A <-> y e. A ) )
10	8, 9	sbie 1779	. 2 |- ( [ y / w ] w e. A <-> y e. A )
11	3, 7, 10	3bitr3i 209	1 |- ( [ y / x ] x e. A <-> y e. A )

Syntax hints:   <-> wb 104   [wsb 1750   e. wcel 2136   F/_wnfc 2294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2296

This theorem is referenced by:  rmo3f  2922