Theorem cbviun 3715

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 2213	. . . 4 |- F/ y z e. B
3		cbviun.2	. . . . 5 |- F/_ x C
4	3	nfcri 2213	. . . 4 |- F/ x z e. C
5		cbviun.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2148	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrex 2574	. . 3 |- ( E. x e. A z e. B <-> E. y e. A z e. C )
8	7	abbii 2194	. 2 |- { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
9		df-iun 3680	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
10		df-iun 3680	. 2 |- U_ y e. A C = { z | E. y e. A z e. C }
11	8, 9, 10	3eqtr4i 2111	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {cab 2067   F/_wnfc 2206   E.wrex 2349   U_ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-iun 3680

This theorem is referenced by:  cbviunv  3717  funiunfvdmf  5424  mpt2mptsx  5843  dmmpt2ssx  5845  fmpt2x  5846