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Theorem cbviun 3722
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 2188 . . . 4  |-  F/ y  z  e.  B
3 cbviun.2 . . . . 5  |-  F/_ x C
43nfcri 2188 . . . 4  |-  F/ x  z  e.  C
5 cbviun.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2123 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrex 2547 . . 3  |-  ( E. x  e.  A  z  e.  B  <->  E. y  e.  A  z  e.  C )
87abbii 2169 . 2  |-  { z  |  E. x  e.  A  z  e.  B }  =  { z  |  E. y  e.  A  z  e.  C }
9 df-iun 3687 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
10 df-iun 3687 . 2  |-  U_ y  e.  A  C  =  { z  |  E. y  e.  A  z  e.  C }
118, 9, 103eqtr4i 2086 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409   {cab 2042   F/_wnfc 2181   E.wrex 2324   U_ciun 3685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-iun 3687
This theorem is referenced by:  cbviunv  3724  funiunfvdmf  5431  mpt2mptsx  5851  dmmpt2ssx  5853  fmpt2x  5854
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