Theorem csbvarg 3108

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 2771	. 2 |- ( A e. V -> A e. _V )
2		vex 2763	. . . . . 6 |- y e. _V
3		df-csb 3081	. . . . . . 7 |- [_ y / x ]_ x = { z | [. y / x ]. z e. x }
4		sbcel2gv 3049	. . . . . . . 8 |- ( y e. _V -> ( [. y / x ]. z e. x <-> z e. y ) )
5	4	abbi1dv 2313	. . . . . . 7 |- ( y e. _V -> { z | [. y / x ]. z e. x } = y )
6	3, 5	eqtrid 2238	. . . . . 6 |- ( y e. _V -> [_ y / x ]_ x = y )
7	2, 6	ax-mp 5	. . . . 5 |- [_ y / x ]_ x = y
8	7	csbeq2i 3107	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / y ]_ y
9		csbco 3090	. . . 4 |- [_ A / y ]_ [_ y / x ]_ x = [_ A / x ]_ x
10		df-csb 3081	. . . 4 |- [_ A / y ]_ y = { z | [. A / y ]. z e. y }
11	8, 9, 10	3eqtr3i 2222	. . 3 |- [_ A / x ]_ x = { z | [. A / y ]. z e. y }
12		sbcel2gv 3049	. . . 4 |- ( A e. _V -> ( [. A / y ]. z e. y <-> z e. A ) )
13	12	abbi1dv 2313	. . 3 |- ( A e. _V -> { z | [. A / y ]. z e. y } = A )
14	11, 13	eqtrid 2238	. 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 1364   e. wcel 2164   {cab 2179   _Vcvv 2760   [.wsbc 2985   [_csb 3080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-csb 3081

This theorem is referenced by:  sbccsb2g  3110  csbfvg  5594  f1od2  6288  csbwrdg  10943  divcncfap  14768  bj-sels  15406