Theorem cbviun 3850

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Hypotheses

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

Assertion

Ref	Expression
cbviun	|- U_ x e. A B = U_ y e. A C

Distinct variable groups:   y, A   x, A

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

Proof of Theorem cbviun

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviun.1	. . . . 5 |- F/_ y B
2	1	nfcri 2275	. . . 4 |- F/ y z e. B
3		cbviun.2	. . . . 5 |- F/_ x C
4	3	nfcri 2275	. . . 4 |- F/ x z e. C
5		cbviun.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2209	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrex 2651	. . 3 |- ( E. x e. A z e. B <-> E. y e. A z e. C )
8	7	abbii 2255	. 2 |- { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
9		df-iun 3815	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
10		df-iun 3815	. 2 |- U_ y e. A C = { z | E. y e. A z e. C }
11	8, 9, 10	3eqtr4i 2170	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   {cab 2125   F/_wnfc 2268   E.wrex 2417   U_ciun 3813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-iun 3815

This theorem is referenced by:  cbviunv  3852  funiunfvdmf  5665  mpomptsx  6095  dmmpossx  6097  fmpox  6098  fsum2dlemstep  11203  fisumcom2  11207  fsumiun  11246  ctiunctlemf  11951