Theorem cbviin 4558

Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
cbviun.1	|- F/_ y B
cbviun.2	|- F/_ x C
cbviun.3	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbviin	|- |^|_ x e. A B = |^|_ y e. A C

Distinct variable groups:   y, A   x, A

Allowed substitution hints:   B( x, y)   C( x, y)

Proof of Theorem cbviin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviun.1	. . . . 5 |- F/_ y B
2	1	nfcri 2758	. . . 4 |- F/ y z e. B
3		cbviun.2	. . . . 5 |- F/_ x C
4	3	nfcri 2758	. . . 4 |- F/ x z e. C
5		cbviun.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2687	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvral 3167	. . 3 |- ( A. x e. A z e. B <-> A. y e. A z e. C )
8	7	abbii 2739	. 2 |- { z | A. x e. A z e. B } = { z | A. y e. A z e. C }
9		df-iin 4523	. 2 |- |^|_ x e. A B = { z | A. x e. A z e. B }
10		df-iin 4523	. 2 |- |^|_ y e. A C = { z | A. y e. A z e. C }
11	8, 9, 10	3eqtr4i 2654	1 |- |^|_ x e. A B = |^|_ y e. A C

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   |^|_ciin 4521

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-ral 2917  df-iin 4523

This theorem is referenced by:  cbviinv  4560  elrfirn2  37259  fnlimfvre  39906  smflimlem6  40984  smflim  40985  smflim2  41012  smfsup  41020  smfinflem  41023  smfinf  41024  smflimsup  41034  smfliminf  41037