Theorem csbvarg 3030

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 2697	. 2 |- ( A e. V -> A e. _V )
2		vex 2689	. . . . . 6 |- y e. _V
3		df-csb 3004	. . . . . . 7 |- [_ y / x ]_ x = { z | [. y / x ]. z e. x }
4		sbcel2gv 2972	. . . . . . . 8 |- ( y e. _V -> ( [. y / x ]. z e. x <-> z e. y ) )
5	4	abbi1dv 2259	. . . . . . 7 |- ( y e. _V -> { z | [. y / x ]. z e. x } = y )
6	3, 5	syl5eq 2184	. . . . . 6 |- ( y e. _V -> [_ y / x ]_ x = y )
7	2, 6	ax-mp 5	. . . . 5 |- [_ y / x ]_ x = y
8	7	csbeq2i 3029	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / y ]_ y
9		csbco 3013	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / x ]_ x
10		df-csb 3004	. . . 4 |- [_ A / y ]_ y = { z | [. A / y ]. z e. y }
11	8, 9, 10	3eqtr3i 2168	. . 3 |- [_ A / x ]_ x = { z | [. A / y ]. z e. y }
12		sbcel2gv 2972	. . . 4 |- ( A e. _V -> ( [. A / y ]. z e. y <-> z e. A ) )
13	12	abbi1dv 2259	. . 3 |- ( A e. _V -> { z | [. A / y ]. z e. y } = A )
14	11, 13	syl5eq 2184	. 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 1331   e. wcel 1480   {cab 2125   _Vcvv 2686   [.wsbc 2909   [_csb 3003

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004

This theorem is referenced by:  sbccsb2g  3032  csbfvg  5459  f1od2  6132  bj-sels  13112