Theorem cbviun 3923

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 2313	. . . 4 |- F/ y z e. B
3		cbviun.2	. . . . 5 |- F/_ x C
4	3	nfcri 2313	. . . 4 |- F/ x z e. C
5		cbviun.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2247	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrex 2700	. . 3 |- ( E. x e. A z e. B <-> E. y e. A z e. C )
8	7	abbii 2293	. 2 |- { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
9		df-iun 3888	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
10		df-iun 3888	. 2 |- U_ y e. A C = { z | E. y e. A z e. C }
11	8, 9, 10	3eqtr4i 2208	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {cab 2163   F/_wnfc 2306   E.wrex 2456   U_ciun 3886

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-iun 3888

This theorem is referenced by:  cbviunv  3925  funiunfvdmf  5761  mpomptsx  6194  dmmpossx  6196  fmpox  6197  fsum2dlemstep  11434  fisumcom2  11438  fsumiun  11477  fprod2dlemstep  11622  fprodcom2fi  11626  ctiunctlemf  12430  ctiunctal  12433