Theorem cbvcsb 3538

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 2758	. . . 4 |- F/ y z e. C
3		cbvcsb.2	. . . . 5 |- F/_ x D
4	3	nfcri 2758	. . . 4 |- F/ x z e. D
5		cbvcsb.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2687	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbc 3464	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2739	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 3534	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 3534	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2654	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-sbc 3436  df-csb 3534

This theorem is referenced by:  cbvcsbv  3539  cbvsum  14425  cbvprod  14645  measiuns  30280  poimirlem26  33435  climinf2mpt  39946  climinfmpt  39947