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Theorem iuneq1d 4007
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 3997 . 2  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
31, 2syl 15 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   U_ciun 3984
This theorem is referenced by:  iuneq12d  4008  disjxiun  4099  kmlem11  7873  prmreclem4  13057  imasval  13507  iundisj  19003  iundisj2  19004  voliunlem1  19005  iunmbl  19008  volsup  19011  uniioombllem4  19039  iuninc  23207  iundisjf  23224  iundisj2f  23225  iundisjfi  23353  iundisj2fi  23354  iundisjcnt  23355  indval2  23678  sigaclcu3  23771  meascnbl  23837  cvmliftlem10  24229  voliunnfl  25490  volsupnfl  25491  heiborlem3  25860  heibor  25868  bnj1113  28562  bnj570  28682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-v 2866  df-in 3235  df-ss 3242  df-iun 3986
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