Theorem cbviun 4044

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 2386	. . . 4 |- F/ y z e. B
3		cbviun.2	. . . . 5 |- F/_ x C
4	3	nfcri 2386	. . . 4 |- F/ x z e. C
5		cbviun.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2308	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrex 2783	. . 3 |- ( E. x e. A z e. B <-> E. y e. A z e. C )
8	7	abbii 2354	. 2 |- { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
9		df-iun 4009	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
10		df-iun 4009	. 2 |- U_ y e. A C = { z | E. y e. A z e. C }
11	8, 9, 10	3eqtr4i 2269	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   F/_wnfc 2379   E.wrex 2529   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-iun 4009

This theorem is referenced by:  cbviunv  4046  funiunfvdmf  5960  mpomptsx  6423  dmmpossx  6425  fmpox  6426  fsum2dlemstep  12179  fisumcom2  12183  fsumiun  12222  fprod2dlemstep  12367  fprodcom2fi  12371  ctiunctlemf  13307  ctiunctal  13310