Theorem csbvarg 2564

Description: The proper substitution of a class for set variable results in the class (if the class exists).

Assertion

Ref	Expression
csbvarg	|- (A e. B -> [_A / x]_x = A)

Proof of Theorem csbvarg

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. B -> A e. _V)
2		visset 2295	. . . . 5 |- y e. _V
3		sbcel2gv 2512	. . . . . . 7 |- (y e. _V -> ([y / x]z e. x <-> z e. y))
4	3	abbi1dv 2010	. . . . . 6 |- (y e. _V -> {z | [y / x]z e. x} = y)
5		df-csb 2541	. . . . . 6 |- [_y / x]_x = {z | [y / x]z e. x}
6	4, 5	syl5eq 1940	. . . . 5 |- (y e. _V -> [_y / x]_x = y)
7	2, 6	ax-mp 7	. . . 4 |- [_y / x]_x = y
8	7	csbeq2i 2563	. . 3 |- (A e. _V -> [_A / y]_[_y / x]_x = [_A / y]_y)
9		csbcog 2547	. . 3 |- (A e. _V -> [_A / y]_[_y / x]_x = [_A / x]_x)
10		sbcel2gv 2512	. . . . 5 |- (A e. _V -> ([A / y]z e. y <-> z e. A))
11	10	abbi1dv 2010	. . . 4 |- (A e. _V -> {z | [A / y]z e. y} = A)
12		df-csb 2541	. . . 4 |- [_A / y]_y = {z | [A / y]z e. y}
13	11, 12	syl5eq 1940	. . 3 |- (A e. _V -> [_A / y]_y = A)
14	8, 9, 13	3eqtr3d 1934	. 2 |- (A e. _V -> [_A / x]_x = A)
15	1, 14	syl 12	1 |- (A e. B -> [_A / x]_x = A)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292  [_csb 2540

This theorem is referenced by:  sbccsb2g 2566  intab 3247  csbfvg 4701  arisumilem 8486  efaddlem5 8604  oprcn 9255  ipval2lem1 9690  kbass2 11688  kbass5 11691  bnj33 12401  dfdir2 14639  rusbcALT 16410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

Copyright terms: Public domain