Theorem csbvarg 3164

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 2868	. 2 |- (A e. V -> A e. _V)
2		vex 2863	. . . . . 6 |- y e. _V
3		df-csb 3138	. . . . . . 7 |- [_y / x]_x = {z | [.y / x ].z e. x}
4		sbcel2gv 3107	. . . . . . . 8 |- (y e. _V -> ( [.y / x ].z e. x <-> z e. y))
5	4	abbi1dv 2470	. . . . . . 7 |- (y e. _V -> {z | [.y / x ].z e. x} = y)
6	3, 5	syl5eq 2397	. . . . . 6 |- (y e. _V -> [_y / x]_x = y)
7	2, 6	ax-mp 5	. . . . 5 |- [_y / x]_x = y
8	7	csbeq2i 3163	. . . 4 |- [_A / y]_[_y / x]_x = [_A / y]_y
9		csbco 3146	. . . 4 |- [_A / y]_[_y / x]_x = [_A / x]_x
10		df-csb 3138	. . . 4 |- [_A / y]_y = {z | [.A / y ].z e. y}
11	8, 9, 10	3eqtr3i 2381	. . 3 |- [_A / x]_x = {z | [.A / y ].z e. y}
12		sbcel2gv 3107	. . . 4 |- (A e. _V -> ( [.A / y ].z e. y <-> z e. A))
13	12	abbi1dv 2470	. . 3 |- (A e. _V -> {z | [.A / y ].z e. y} = A)
14	11, 13	syl5eq 2397	. 2 |- (A e. _V -> [_A / x]_x = A)
15	1, 14	syl 15	1 |- (A e. V -> [_A / x]_x = A)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339  _Vcvv 2860   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  sbccsb2g  3166  csbfvg  5339