Theorem cbvcsb 3097

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 2341	. . . 4 |- F/ y z e. C
3		cbvcsb.2	. . . . 5 |- F/_ x D
4	3	nfcri 2341	. . . 4 |- F/ x z e. D
5		cbvcsb.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2274	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbc 3026	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2320	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 3093	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 3093	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2235	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   {cab 2190   F/_wnfc 2334   [.wsbc 2997   [_csb 3092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-sbc 2998  df-csb 3093

This theorem is referenced by:  cbvcsbv  3098  cbvsum  11613