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Theorem cbviun 3910
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 2306 . . . 4  |-  F/ y  z  e.  B
3 cbviun.2 . . . . 5  |-  F/_ x C
43nfcri 2306 . . . 4  |-  F/ x  z  e.  C
5 cbviun.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2240 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrex 2693 . . 3  |-  ( E. x  e.  A  z  e.  B  <->  E. y  e.  A  z  e.  C )
87abbii 2286 . 2  |-  { z  |  E. x  e.  A  z  e.  B }  =  { z  |  E. y  e.  A  z  e.  C }
9 df-iun 3875 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
10 df-iun 3875 . 2  |-  U_ y  e.  A  C  =  { z  |  E. y  e.  A  z  e.  C }
118, 9, 103eqtr4i 2201 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   {cab 2156   F/_wnfc 2299   E.wrex 2449   U_ciun 3873
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-iun 3875
This theorem is referenced by:  cbviunv  3912  funiunfvdmf  5743  mpomptsx  6176  dmmpossx  6178  fmpox  6179  fsum2dlemstep  11397  fisumcom2  11401  fsumiun  11440  fprod2dlemstep  11585  fprodcom2fi  11589  ctiunctlemf  12393  ctiunctal  12396
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