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Theorem iuneq2d 3930
Description: Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
Hypothesis
Ref Expression
iuneq2d.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
iuneq2d  |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
Distinct variable groups:    ph, x    x, A
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem iuneq2d
StepHypRef Expression
1 iuneq2d.2 . . 3  |-  ( ph  ->  B  =  C )
21adantr 451 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32iuneq2dv 3926 1  |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   U_ciun 3905
This theorem is referenced by:  iununi  3986  oelim2  6593  ituniiun  8048  imasval  13414  mreacs  13560  taylfval  19738  dfrtrclrec2  24040  rtrclreclem.refl  24041  rtrclreclem.subset  24042  rtrclreclem.min  24044  trpredeq1  24223  trpredeq2  24224  trclval  25894  isKleene  25988  neibastop2  26310  sstotbnd2  26498  equivtotbnd  26502  totbndbnd  26513  heiborlem3  26537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iun 3907
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