Theorem clelsb3f 2286

Description: Substitution applied to an atomic wff (class version of elsb3 1952). (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
clelsb3f.1	|- F/_ x A

Assertion

Ref	Expression
clelsb3f	|- ( [ y / x ] x e. A <-> y e. A )

Proof of Theorem clelsb3f

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb3f.1	. . . 4 |- F/_ x A
2	1	nfcri 2276	. . 3 |- F/ x w e. A
3	2	sbco2 1939	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / w ] w e. A )
4		nfv 1509	. . . 4 |- F/ w x e. A
5		eleq1w 2201	. . . 4 |- ( w = x -> ( w e. A <-> x e. A ) )
6	4, 5	sbie 1765	. . 3 |- ( [ x / w ] w e. A <-> x e. A )
7	6	sbbii 1739	. 2 |- ( [ y / x ] [ x / w ] w e. A <-> [ y / x ] x e. A )
8		nfv 1509	. . 3 |- F/ w y e. A
9		eleq1w 2201	. . 3 |- ( w = y -> ( w e. A <-> y e. A ) )
10	8, 9	sbie 1765	. 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   e. wcel 1481   [wsb 1736   F/_wnfc 2269

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271

This theorem is referenced by:  rmo3f  2885