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Theorem iuneq2dv 4106
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 2781 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 iuneq2 4101 . 2  |-  ( A. x  e.  A  B  =  C  ->  U_ x  e.  A  B  =  U_ x  e.  A  C
)
42, 3syl 16 1  |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   U_ciun 4085
This theorem is referenced by:  iuneq12d  4109  iuneq2d  4110  fparlem3  6440  fparlem4  6441  oalim  6768  omlim  6769  oelim  6770  oelim2  6830  r1val3  7756  imasdsval  13733  acsfn  13876  tgidm  17037  cmpsub  17455  alexsublem  18067  bcth3  19276  ovoliunlem1  19390  voliunlem1  19436  uniiccdif  19462  uniioombllem2  19467  uniioombllem3a  19468  uniioombllem4  19470  itg2monolem1  19634  taylpfval  20273  cvmscld  24952  mblfinlem  26234  ftc1anclem6  26275  heibor  26521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-iun 4087
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