MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iuneq2dv Unicode version

Theorem iuneq2dv 3942
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 2639 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 iuneq2 3937 . 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 1632    e. wcel 1696   A.wral 2556   U_ciun 3921
This theorem is referenced by:  iuneq12d  3945  iuneq2d  3946  fparlem3  6236  fparlem4  6237  oalim  6547  omlim  6548  oelim  6549  oelim2  6609  r1val3  7526  imasdsval  13434  acsfn  13577  tgidm  16734  cmpsub  17143  alexsublem  17754  bcth3  18769  ovoliunlem1  18877  voliunlem1  18923  uniiccdif  18949  uniioombllem2  18954  uniioombllem3a  18955  uniioombllem4  18957  itg2monolem1  19121  taylpfval  19760  cvmscld  23819  imfstnrelc  25184  heibor  26648
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-v 2803  df-in 3172  df-ss 3179  df-iun 3923
  Copyright terms: Public domain W3C validator