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Theorem cbviunv 3957
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
cbviunv.1  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbviunv  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Distinct variable groups:    x, A    y, A    y, B    x, C
Allowed substitution hints:    B( x)    C( y)

Proof of Theorem cbviunv
StepHypRef Expression
1 nfcv 2432 . 2  |-  F/_ y B
2 nfcv 2432 . 2  |-  F/_ x C
3 cbviunv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbviun 3955 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   U_ciun 3921
This theorem is referenced by:  iunxdif2  3966  onfununi  6374  oelim2  6609  marypha2lem2  7205  trcl  7426  r1om  7886  fictb  7887  cfsmolem  7912  cfsmo  7913  domtriomlem  8084  domtriom  8085  pwfseq  8302  wunex2  8376  wuncval2  8385  ackbijnn  12302  efgs1b  15061  ablfaclem3  15338  ptbasfi  17292  bcth3  18769  itg1climres  19085  cvmliftlem15  23844  trpredlem1  24301  trpredtr  24304  trpredmintr  24305  trpredrec  24312  clscnc  26113  neibastop2  26413  filnetlem4  26433  sstotbnd2  26601  heiborlem3  26640  heibor  26648  bnj601  29268  lcfr  32397  mapdrval  32459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-iun 3923
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