Theorem csbvarg 2018

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 1814	. 2 |- (A e. B -> A e. V)
2		visset 1810	. . . . 5 |- y e. V
3		sbcel2gv 1978	. . . . . . 7 |- (y e. V -> ([y / x]z e. x <-> z e. y))
4	3	abbi1dv 1577	. . . . . 6 |- (y e. V -> {z | [y / x]z e. x} = y)
5		df-csb 1999	. . . . . 6 |- [_y / x]_x = {z | [y / x]z e. x}
6	4, 5	syl5eq 1517	. . . . 5 |- (y e. V -> [_y / x]_x = y)
7	2, 6	ax-mp 7	. . . 4 |- [_y / x]_x = y
8	7	csbeq2i 2017	. . 3 |- (A e. V -> [_A / y]_[_y / x]_x = [_A / y]_y)
9		csbcog 2004	. . 3 |- (A e. V -> [_A / y]_[_y / x]_x = [_A / x]_x)
10		sbcel2gv 1978	. . . . 5 |- (A e. V -> ([A / y]z e. y <-> z e. A))
11	10	abbi1dv 1577	. . . 4 |- (A e. V -> {z | [A / y]z e. y} = A)
12		df-csb 1999	. . . 4 |- [_A / y]_y = {z | [A / y]z e. y}
13	11, 12	syl5eq 1517	. . 3 |- (A e. V -> [_A / y]_y = A)
14	8, 9, 13	3eqtr3d 1513	. 2 |- (A e. V -> [_A / x]_x = A)
15	1, 14	syl 10	1 |- (A e. B -> [_A / x]_x = A)

Syntax hints:   -> wi 3   = wceq 955   e. wcel 957  [wsbc 1169  {cab 1462  Vcvv 1808  [_csb 1998

This theorem is referenced by:  sbccsb2g 2020  intab 2556  csbfvg 3739  fnsmntlem 7177  efaddlem5 7301  oprcn 7939  ipval2lem1 8313  kbass2t 10006  kbass5t 10009

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939  df-csb 1999

Copyright terms: Public domain