Theorem csbvarg 2933

Description: The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbvarg	|- ( A e. V -> [_ A / x ]_ x = A )

Proof of Theorem csbvarg

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2610	. 2 |- ( A e. V -> A e. _V )
2		vex 2604	. . . . . 6 |- y e. _V
3		df-csb 2909	. . . . . . 7 |- [_ y / x ]_ x = { z | [. y / x ]. z e. x }
4		sbcel2gv 2877	. . . . . . . 8 |- ( y e. _V -> ( [. y / x ]. z e. x <-> z e. y ) )
5	4	abbi1dv 2198	. . . . . . 7 |- ( y e. _V -> { z | [. y / x ]. z e. x } = y )
6	3, 5	syl5eq 2125	. . . . . 6 |- ( y e. _V -> [_ y / x ]_ x = y )
7	2, 6	ax-mp 7	. . . . 5 |- [_ y / x ]_ x = y
8	7	csbeq2i 2932	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / y ]_ y
9		csbco 2917	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / x ]_ x
10		df-csb 2909	. . . 4 |- [_ A / y ]_ y = { z | [. A / y ]. z e. y }
11	8, 9, 10	3eqtr3i 2109	. . 3 |- [_ A / x ]_ x = { z | [. A / y ]. z e. y }
12		sbcel2gv 2877	. . . 4 |- ( A e. _V -> ( [. A / y ]. z e. y <-> z e. A ) )
13	12	abbi1dv 2198	. . 3 |- ( A e. _V -> { z | [. A / y ]. z e. y } = A )
14	11, 13	syl5eq 2125	. 2 |- ( A e. _V -> [_ A / x ]_ x = A )
15	1, 14	syl 14	1 |- ( A e. V -> [_ A / x ]_ x = A )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {cab 2067   _Vcvv 2601   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  sbccsb2g  2935  csbfvg  5232  f1od2  5876  bj-sels  10705