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Theorem iuneq2dv 3926
Description: Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
Hypothesis
Ref Expression
iuneq2dv.1  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
iuneq2dv  |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem iuneq2dv
StepHypRef Expression
1 iuneq2dv.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
21ralrimiva 2626 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 iuneq2 3921 . 2  |-  ( A. x  e.  A  B  =  C  ->  U_ x  e.  A  B  =  U_ x  e.  A  C
)
42, 3syl 15 1  |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   U_ciun 3905
This theorem is referenced by:  iuneq12d  3929  iuneq2d  3930  fparlem3  6220  fparlem4  6221  oalim  6531  omlim  6532  oelim  6533  oelim2  6593  r1val3  7510  imasdsval  13418  acsfn  13561  tgidm  16718  cmpsub  17127  alexsublem  17738  bcth3  18753  ovoliunlem1  18861  voliunlem1  18907  uniiccdif  18933  uniioombllem2  18938  uniioombllem3a  18939  uniioombllem4  18941  itg2monolem1  19105  taylpfval  19744  cvmscld  23804  imfstnrelc  25081  heibor  26545
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-v 2790  df-in 3159  df-ss 3166  df-iun 3907
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