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Theorem iuneq2d 4120
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 453 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32iuneq2dv 4116 1  |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   U_ciun 4095
This theorem is referenced by:  iununi  4178  oelim2  6841  ituniiun  8307  imasval  13742  mreacs  13888  cnextval  18097  taylfval  20280  dfrtrclrec2  25148  rtrclreclem.refl  25149  rtrclreclem.subset  25150  rtrclreclem.min  25152  trpredeq1  25503  trpredeq2  25504  voliunnfl  26262  neibastop2  26404  sstotbnd2  26497  equivtotbnd  26501  totbndbnd  26512  heiborlem3  26536  otiunsndisjX  28082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-in 3329  df-ss 3336  df-iun 4097
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