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Theorem seqeq3d 11924
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
Hypothesis
Ref Expression
seqeqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
seqeq3d  |-  ( ph  ->  seq M (  .+  ,  A )  =  seq M (  .+  ,  B ) )

Proof of Theorem seqeq3d
StepHypRef Expression
1 seqeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 seqeq3 11921 . 2  |-  ( A  =  B  ->  seq M (  .+  ,  A )  =  seq M (  .+  ,  B ) )
31, 2syl 16 1  |-  ( ph  ->  seq M (  .+  ,  A )  =  seq M (  .+  ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    seqcseq 11916
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-or 370  df-an 371  df-3an 967  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-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-cnv 4949  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-recs 6935  df-rdg 6969  df-seq 11917
This theorem is referenced by:  seqeq123d  11925  seqf1olem2  11956  seqf1o  11957  seqof2  11974  expval  11977  sumeq1  13277  sumeq2w  13280  cbvsum  13283  summo  13305  fsum  13308  geomulcvg  13447  gsumvalx  15613  mulgval  15740  gsumval3eu  16494  gsumval3OLD  16495  gsumval3lem2  16497  gsumzres  16501  gsumzf1o  16504  gsumzresOLD  16505  gsumzf1oOLD  16507  elovolmr  21084  ovolctb  21098  ovoliunlem3  21112  ovoliunnul  21115  ovolshftlem1  21117  voliunlem3  21159  voliun  21161  uniioombllem2  21189  vitalilem4  21217  vitalilem5  21218  dvnfval  21522  mtestbdd  21996  radcnv0  22007  radcnvlt1  22009  radcnvle  22011  psercn  22017  pserdvlem2  22019  abelthlem1  22022  abelthlem3  22024  logtayl  22231  atantayl2  22459  atantayl3  22460  lgsval  22765  lgsval4  22781  lgsneg  22784  lgsmod  22786  dchrmusumlema  22868  dchrisum0lema  22889  gxval  23890  lgamgulm2  27159  lgamcvglem  27163  prodeq1f  27558  prodeq2w  27562  prodmo  27586  fprod  27591  faclim  27689  ovoliunnfl  28574  voliunnfl  28576  stirlinglem5  30014
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