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Theorem iuneq2d 3946
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 3942 1  |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   U_ciun 3921
This theorem is referenced by:  iununi  4002  oelim2  6609  ituniiun  8064  imasval  13430  mreacs  13576  taylfval  19754  dfrtrclrec2  24055  rtrclreclem.refl  24056  rtrclreclem.subset  24057  rtrclreclem.min  24059  trpredeq1  24294  trpredeq2  24295  trclval  25997  isKleene  26091  neibastop2  26413  sstotbnd2  26601  equivtotbnd  26605  totbndbnd  26616  heiborlem3  26640
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
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