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Theorem cbviun 3886
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.
StepHypRef Expression
1 cbviun.1 . . . . 5  |-  F/_ y B
21nfcri 2293 . . . 4  |-  F/ y  z  e.  B
3 cbviun.2 . . . . 5  |-  F/_ x C
43nfcri 2293 . . . 4  |-  F/ x  z  e.  C
5 cbviun.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2227 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrex 2677 . . 3  |-  ( E. x  e.  A  z  e.  B  <->  E. y  e.  A  z  e.  C )
87abbii 2273 . 2  |-  { z  |  E. x  e.  A  z  e.  B }  =  { z  |  E. y  e.  A  z  e.  C }
9 df-iun 3851 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
10 df-iun 3851 . 2  |-  U_ y  e.  A  C  =  { z  |  E. y  e.  A  z  e.  C }
118, 9, 103eqtr4i 2188 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128   {cab 2143   F/_wnfc 2286   E.wrex 2436   U_ciun 3849
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-iun 3851
This theorem is referenced by:  cbviunv  3888  funiunfvdmf  5714  mpomptsx  6145  dmmpossx  6147  fmpox  6148  fsum2dlemstep  11331  fisumcom2  11335  fsumiun  11374  fprod2dlemstep  11519  fprodcom2fi  11523  ctiunctlemf  12167  ctiunctal  12170
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