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Theorem esumeq2dv 26492
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Hypothesis
Ref Expression
esumeq2dv.1  |-  ( (
ph  /\  k  e.  A )  ->  B  =  C )
Assertion
Ref Expression
esumeq2dv  |-  ( ph  -> Σ* k  e.  A B  = Σ* k  e.  A C )
Distinct variable group:    ph, k
Allowed substitution hints:    A( k)    B( k)    C( k)

Proof of Theorem esumeq2dv
StepHypRef Expression
1 nfv 1673 . 2  |-  F/ k
ph
2 esumeq2dv.1 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  =  C )
32ralrimiva 2797 . 2  |-  ( ph  ->  A. k  e.  A  B  =  C )
41, 3esumeq2d 26491 1  |-  ( ph  -> Σ* k  e.  A B  = Σ* k  e.  A C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756  Σ*cesum 26481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-iota 5379  df-fv 5424  df-ov 6092  df-esum 26482
This theorem is referenced by:  esumeq2sdv  26493  esumle  26506  esummulc1  26528  esummulc2  26529  esumdivc  26530  measinb  26633  measres  26634  measdivcstOLD  26636  measdivcst  26637  cntmeas  26638  omsval  26706  totprobd  26807
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