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Theorem cbviunv 3941
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 2419 . 2  |-  F/_ y B
2 nfcv 2419 . 2  |-  F/_ x C
3 cbviunv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbviun 3939 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   U_ciun 3905
This theorem is referenced by:  iunxdif2  3950  onfununi  6358  oelim2  6593  marypha2lem2  7189  trcl  7410  r1om  7870  fictb  7871  cfsmolem  7896  cfsmo  7897  domtriomlem  8068  domtriom  8069  pwfseq  8286  wunex2  8360  wuncval2  8369  ackbijnn  12286  efgs1b  15045  ablfaclem3  15322  ptbasfi  17276  bcth3  18753  itg1climres  19069  cvmliftlem15  23829  trpredlem1  24230  trpredtr  24233  trpredmintr  24234  trpredrec  24241  clscnc  26010  neibastop2  26310  filnetlem4  26330  sstotbnd2  26498  heiborlem3  26537  heibor  26545  bnj601  28952  lcfr  31775  mapdrval  31837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-iun 3907
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