Theorem csbvarg 3073

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 2737	. 2 |- ( A e. V -> A e. _V )
2		vex 2729	. . . . . 6 |- y e. _V
3		df-csb 3046	. . . . . . 7 |- [_ y / x ]_ x = { z | [. y / x ]. z e. x }
4		sbcel2gv 3014	. . . . . . . 8 |- ( y e. _V -> ( [. y / x ]. z e. x <-> z e. y ) )
5	4	abbi1dv 2286	. . . . . . 7 |- ( y e. _V -> { z | [. y / x ]. z e. x } = y )
6	3, 5	syl5eq 2211	. . . . . 6 |- ( y e. _V -> [_ y / x ]_ x = y )
7	2, 6	ax-mp 5	. . . . 5 |- [_ y / x ]_ x = y
8	7	csbeq2i 3072	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / y ]_ y
9		csbco 3055	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / x ]_ x
10		df-csb 3046	. . . 4 |- [_ A / y ]_ y = { z | [. A / y ]. z e. y }
11	8, 9, 10	3eqtr3i 2194	. . 3 |- [_ A / x ]_ x = { z | [. A / y ]. z e. y }
12		sbcel2gv 3014	. . . 4 |- ( A e. _V -> ( [. A / y ]. z e. y <-> z e. A ) )
13	12	abbi1dv 2286	. . 3 |- ( A e. _V -> { z | [. A / y ]. z e. y } = A )
14	11, 13	syl5eq 2211	. 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 1343   e. wcel 2136   {cab 2151   _Vcvv 2726   [.wsbc 2951   [_csb 3045

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  sbccsb2g  3075  csbfvg  5524  f1od2  6203  bj-sels  13796