Theorem cbvcsb 2912

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 2213	. . . 4 |- F/ y z e. C
3		cbvcsb.2	. . . . 5 |- F/_ x D
4	3	nfcri 2213	. . . 4 |- F/ x z e. D
5		cbvcsb.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2148	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbc 2842	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2194	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 2909	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 2909	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2111	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {cab 2067   F/_wnfc 2206   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-sbc 2816  df-csb 2909

This theorem is referenced by:  cbvcsbv  2913