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

Theorem iuneq1d 4350
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 4339 . 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 1379   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-in 3483  df-ss 3490  df-iun 4327
This theorem is referenced by:  iuneq12d  4351  disjxiun  4444  kmlem11  8536  prmreclem4  14292  imasval  14762  iundisj  21693  iundisj2  21694  voliunlem1  21695  iunmbl  21698  volsup  21701  uniioombllem4  21730  iuninc  27101  iundisjf  27121  iundisj2f  27122  iundisjfi  27269  iundisj2fi  27270  iundisjcnt  27271  indval2  27668  sigaclcu3  27762  meascnbl  27830  cvmliftlem10  28379  voliunnfl  29635  volsupnfl  29636  heiborlem3  29912  heibor  29920  bnj1113  32923  bnj155  33016  bnj570  33042  bnj893  33065
  Copyright terms: Public domain W3C validator