Theorem csbvarg 2994

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 2666	. 2 |- ( A e. V -> A e. _V )
2		vex 2658	. . . . . 6 |- y e. _V
3		df-csb 2970	. . . . . . 7 |- [_ y / x ]_ x = { z | [. y / x ]. z e. x }
4		sbcel2gv 2938	. . . . . . . 8 |- ( y e. _V -> ( [. y / x ]. z e. x <-> z e. y ) )
5	4	abbi1dv 2232	. . . . . . 7 |- ( y e. _V -> { z | [. y / x ]. z e. x } = y )
6	3, 5	syl5eq 2157	. . . . . 6 |- ( y e. _V -> [_ y / x ]_ x = y )
7	2, 6	ax-mp 7	. . . . 5 |- [_ y / x ]_ x = y
8	7	csbeq2i 2993	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / y ]_ y
9		csbco 2978	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / x ]_ x
10		df-csb 2970	. . . 4 |- [_ A / y ]_ y = { z | [. A / y ]. z e. y }
11	8, 9, 10	3eqtr3i 2141	. . 3 |- [_ A / x ]_ x = { z | [. A / y ]. z e. y }
12		sbcel2gv 2938	. . . 4 |- ( A e. _V -> ( [. A / y ]. z e. y <-> z e. A ) )
13	12	abbi1dv 2232	. . 3 |- ( A e. _V -> { z | [. A / y ]. z e. y } = A )
14	11, 13	syl5eq 2157	. 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 1312   e. wcel 1461   {cab 2099   _Vcvv 2655   [.wsbc 2876   [_csb 2969

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-sbc 2877  df-csb 2970

This theorem is referenced by:  sbccsb2g  2996  csbfvg  5411  f1od2  6084  bj-sels  12795