Theorem clelsb3f 2768

Description: Substitution applied to an atomic wff (class version of elsb3 2434). (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/_ y A

Assertion

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

Proof of Theorem clelsb3f

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb3f.1	. . . 4 |- F/_ y A
2	1	nfcri 2758	. . 3 |- F/ y w e. A
3	2	sbco2 2415	. 2 |- ( [ x / y ] [ y / w ] w e. A <-> [ x / w ] w e. A )
4		nfv 1843	. . . 4 |- F/ w y e. A
5		eleq1 2689	. . . 4 |- ( w = y -> ( w e. A <-> y e. A ) )
6	4, 5	sbie 2408	. . 3 |- ( [ y / w ] w e. A <-> y e. A )
7	6	sbbii 1887	. 2 |- ( [ x / y ] [ y / w ] w e. A <-> [ x / y ] y e. A )
8		nfv 1843	. . 3 |- F/ w x e. A
9		eleq1 2689	. . 3 |- ( w = x -> ( w e. A <-> x e. A ) )
10	8, 9	sbie 2408	. 2 |- ( [ x / w ] w e. A <-> x e. A )
11	3, 7, 10	3bitr3i 290	1 |- ( [ x / y ] y e. A <-> x e. A )

Syntax hints:   <-> wb 196   [wsb 1880   e. wcel 1990   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  rmo3f  29335  suppss2f  29439  fmptdF  29456  disjdsct  29480  esumpfinvalf  30138