Theorem cbviun 3694

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 2172	. . . 4 |- F/ y z e. B
3		cbviun.2	. . . . 5 |- F/_ x C
4	3	nfcri 2172	. . . 4 |- F/ x z e. C
5		cbviun.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2107	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrex 2530	. . 3 |- ( E. x e. A z e. B <-> E. y e. A z e. C )
8	7	abbii 2153	. 2 |- { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
9		df-iun 3659	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
10		df-iun 3659	. 2 |- U_ y e. A C = { z | E. y e. A z e. C }
11	8, 9, 10	3eqtr4i 2070	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   {cab 2026   F/_wnfc 2165   E.wrex 2307   U_ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-iun 3659

This theorem is referenced by:  cbviunv  3696  funiunfvdmf  5403  mpt2mptsx  5823  dmmpt2ssx  5825  fmpt2x  5826