Theorem cbviin 4003

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 2366	. . . 4 |- F/ y z e. B
3		cbviun.2	. . . . 5 |- F/_ x C
4	3	nfcri 2366	. . . 4 |- F/ x z e. C
5		cbviun.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2299	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvral 2761	. . 3 |- ( A. x e. A z e. B <-> A. y e. A z e. C )
8	7	abbii 2345	. 2 |- { z | A. x e. A z e. B } = { z | A. y e. A z e. C }
9		df-iin 3968	. 2 |- |^|_ x e. A B = { z | A. x e. A z e. B }
10		df-iin 3968	. 2 |- |^|_ y e. A C = { z | A. y e. A z e. C }
11	8, 9, 10	3eqtr4i 2260	1 |- |^|_ x e. A B = |^|_ y e. A C

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {cab 2215   F/_wnfc 2359   A.wral 2508   |^|_ciin 3966

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-iin 3968

This theorem is referenced by:  cbviinv  4005