Theorem cbvcsb 3089

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 2333	. . . 4 |- F/ y z e. C
3		cbvcsb.2	. . . . 5 |- F/_ x D
4	3	nfcri 2333	. . . 4 |- F/ x z e. D
5		cbvcsb.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2266	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbc 3018	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2312	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 3085	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 3085	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2227	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   {cab 2182   F/_wnfc 2326   [.wsbc 2989   [_csb 3084

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-sbc 2990  df-csb 3085

This theorem is referenced by:  cbvcsbv  3090  cbvsum  11525