Theorem csbvarg 2021

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 1817	. 2 |- (A e. B -> A e. V)
2		visset 1813	. . . . 5 |- y e. V
3		sbcel2gv 1981	. . . . . . 7 |- (y e. V -> ([y / x]z e. x <-> z e. y))
4	3	abbi1dv 1579	. . . . . 6 |- (y e. V -> {z | [y / x]z e. x} = y)
5		df-csb 2002	. . . . . 6 |- [_y / x]_x = {z | [y / x]z e. x}
6	4, 5	syl5eq 1519	. . . . 5 |- (y e. V -> [_y / x]_x = y)
7	2, 6	ax-mp 7	. . . 4 |- [_y / x]_x = y
8	7	csbeq2i 2020	. . 3 |- (A e. V -> [_A / y]_[_y / x]_x = [_A / y]_y)
9		csbcog 2007	. . 3 |- (A e. V -> [_A / y]_[_y / x]_x = [_A / x]_x)
10		sbcel2gv 1981	. . . . 5 |- (A e. V -> ([A / y]z e. y <-> z e. A))
11	10	abbi1dv 1579	. . . 4 |- (A e. V -> {z | [A / y]z e. y} = A)
12		df-csb 2002	. . . 4 |- [_A / y]_y = {z | [A / y]z e. y}
13	11, 12	syl5eq 1519	. . 3 |- (A e. V -> [_A / y]_y = A)
14	8, 9, 13	3eqtr3d 1515	. 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 956   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811  [_csb 2001

This theorem is referenced by:  sbccsb2g 2023  intab 2560  csbfvg 3744  fnsmntlem 7225  efaddlem5 7342  oprcn 7977  ipval2lem1 8351  kbass2t 10050  kbass5t 10053

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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