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Theorem iuneq2d 4105
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 452 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32iuneq2dv 4101 1  |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   U_ciun 4080
This theorem is referenced by:  iununi  4162  oelim2  6824  ituniiun  8286  imasval  13720  mreacs  13866  cnextval  18075  taylfval  20258  dfrtrclrec2  25126  rtrclreclem.refl  25127  rtrclreclem.subset  25128  rtrclreclem.min  25130  trpredeq1  25473  trpredeq2  25474  voliunnfl  26191  neibastop2  26322  sstotbnd2  26415  equivtotbnd  26419  totbndbnd  26430  heiborlem3  26454  otiunsndisjX  27997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ral 2697  df-rex 2698  df-v 2945  df-in 3314  df-ss 3321  df-iun 4082
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