MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iuneq1d Structured version   Unicode version

Theorem iuneq1d 4340
Description: Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
Hypothesis
Ref Expression
iuneq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
iuneq1d  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)

Proof of Theorem iuneq1d
StepHypRef Expression
1 iuneq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 iuneq1 4329 . 2  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
31, 2syl 16 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383   U_ciun 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-v 3097  df-in 3468  df-ss 3475  df-iun 4317
This theorem is referenced by:  iuneq12d  4341  disjxiun  4434  kmlem11  8543  prmreclem4  14314  imasval  14785  iundisj  21831  iundisj2  21832  voliunlem1  21833  iunmbl  21836  volsup  21839  uniioombllem4  21868  iuninc  27300  iundisjf  27320  iundisj2f  27321  iundisjfi  27473  iundisj2fi  27474  iundisjcnt  27475  indval2  27901  sigaclcu3  27995  meascnbl  28063  cvmliftlem10  28612  mrsubvrs  28755  msubvrs  28793  voliunnfl  30033  volsupnfl  30034  heiborlem3  30284  heibor  30292  bnj1113  33577  bnj155  33670  bnj570  33696  bnj893  33719
  Copyright terms: Public domain W3C validator