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Theorem seqeq3d 10483
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 10480 . 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 10475
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 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-recs 6329  df-frec 6415  df-seqfrec 10476
This theorem is referenced by:  seqeq123d  10484  seq3f1olemstep  10531  seq3f1olemp  10532  exp3val  10552  sumeq1  11394  sumeq2  11398  summodc  11422  zsumdc  11423  fsum3  11426  isumz  11428  prodeq1f  11591  prodeq2w  11595  prodeq2  11596  prodmodc  11617  zproddc  11618  fprodseq  11622  prod1dc  11625  mulgval  13061  lgsval  14858  lgsval4  14874  lgsneg  14878  lgsmod  14880
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