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Theorem iuneq12d 4297
Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
Hypotheses
Ref Expression
iuneq1d.1  |-  ( ph  ->  A  =  B )
iuneq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
iuneq12d  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    C( x)    D( x)

Proof of Theorem iuneq12d
StepHypRef Expression
1 iuneq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21iuneq1d 4296 . 2  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
3 iuneq12d.2 . . . 4  |-  ( ph  ->  C  =  D )
43adantr 465 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  C  =  D )
54iuneq2dv 4293 . 2  |-  ( ph  ->  U_ x  e.  B  C  =  U_ x  e.  B  D )
62, 5eqtrd 2492 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   U_ciun 4272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-v 3073  df-in 3436  df-ss 3443  df-iun 4274
This theorem is referenced by:  cfsmolem  8543  cfsmo  8544  wunex2  9009  wuncval2  9018  imasval  14560  lpival  17442  cnextval  19758  cnextfval  19759  dvfval  21498  mblfinlem2  28570  heiborlem10  28860  otiunsndisj  30273  2spotiundisj  30796
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