Theorem clelsb1f 2353

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2184). (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 2343	. . 3 |- F/ x w e. A
3	2	sbco2 1994	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
4		nfv 1552	. . . 4 |- F/ w x e. A
5		eleq1w 2267	. . . 4 |- ( w = x -> ( w e. A <-> x e. A ) )
6	4, 5	sbie 1815	. . 3 |- ( [ x / w ] w e. A <-> x e. A )
7	6	sbbii 1789	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
8		nfv 1552	. . 3 |- F/ w y e. A
9		eleq1w 2267	. . 3 |- ( w = y -> ( w e. A <-> y e. A ) )
10	8, 9	sbie 1815	. 2 |- ( [ y / w ] w e. A <-> y e. A )
11	3, 7, 10	3bitr3i 210	1 |- ( [ y / x ] x e. A <-> y e. A )

Syntax hints:   <-> wb 105   [wsb 1786   e. wcel 2177   F/_wnfc 2336

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338

This theorem is referenced by:  rmo3f  2974