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Theorem iuneq12d 4351
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 4350 . 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 4347 . 2  |-  ( ph  ->  U_ x  e.  B  C  =  U_ x  e.  B  D )
62, 5eqtrd 2508 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-in 3483  df-ss 3490  df-iun 4327
This theorem is referenced by:  otiunsndisj  4753  cfsmolem  8646  cfsmo  8647  wunex2  9112  wuncval2  9121  imasval  14762  lpival  17675  cnextval  20296  cnextfval  20297  dvfval  22036  2spotiundisj  24739  mblfinlem2  29629  heiborlem10  29919
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