Theorem cbvcsb 3141

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/_yC
cbvcsb.2	|- F/_xD
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/_yC
2	1	nfcri 2484	. . . 4 |- F/y z e. C
3		cbvcsb.2	. . . . 5 |- F/_xD
4	3	nfcri 2484	. . . 4 |- F/x z e. D
5		cbvcsb.3	. . . . 5 |- (x = y -> C = D)
6	5	eleq2d 2420	. . . 4 |- (x = y -> (z e. C <-> z e. D))
7	2, 4, 6	cbvsbc 3075	. . 3 |- ( [.A / x ].z e. C <-> [.A / y ].z e. D)
8	7	abbii 2466	. 2 |- {z | [.A / x ].z e. C} = {z | [.A / y ].z e. D}
9		df-csb 3138	. 2 |- [_A / x]_C = {z | [.A / x ].z e. C}
10		df-csb 3138	. 2 |- [_A / y]_D = {z | [.A / y ].z e. D}
11	8, 9, 10	3eqtr4i 2383	1 |- [_A / x]_C = [_A / y]_D

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2477   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sbc 3048  df-csb 3138

This theorem is referenced by:  cbvcsbv  3142