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Theorem seqeq3d 10516
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 10513 . 2  |-  ( A  =  B  ->  seq M (  .+  ,  A )  =  seq M (  .+  ,  B ) )
31, 2syl 14 1  |-  ( ph  ->  seq M (  .+  ,  A )  =  seq M (  .+  ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    seqcseq 10508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-iota 5207  df-fv 5254  df-ov 5913  df-oprab 5914  df-mpo 5915  df-recs 6349  df-frec 6435  df-seqfrec 10509
This theorem is referenced by:  seqeq123d  10517  seq3f1olemstep  10575  seq3f1olemp  10576  seqf1oglem2  10581  seqf1og  10582  exp3val  10602  sumeq1  11488  sumeq2  11492  summodc  11516  zsumdc  11517  fsum3  11520  isumz  11522  prodeq1f  11685  prodeq2w  11689  prodeq2  11690  prodmodc  11711  zproddc  11712  fprodseq  11716  prod1dc  11719  mulgval  13182  lgsval  15068  lgsval4  15084  lgsneg  15088  lgsmod  15090
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