Theorem cbvcsb 2975

Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
cbvcsb.1	|- F/_ y C
cbvcsb.2	|- F/_ x D
cbvcsb.3	|- ( x = y -> C = D )

Assertion

Ref	Expression
cbvcsb	|- [_ A / x ]_ C = [_ A / y ]_ D

Proof of Theorem cbvcsb

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsb.1	. . . . 5 |- F/_ y C
2	1	nfcri 2249	. . . 4 |- F/ y z e. C
3		cbvcsb.2	. . . . 5 |- F/_ x D
4	3	nfcri 2249	. . . 4 |- F/ x z e. D
5		cbvcsb.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2184	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbc 2905	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2230	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 2972	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 2972	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2145	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1314   e. wcel 1463   {cab 2101   F/_wnfc 2242   [.wsbc 2878   [_csb 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-sbc 2879  df-csb 2972

This theorem is referenced by:  cbvcsbv  2976  cbvsum  11021