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Theorem seqeq3d 10455
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 10452 . 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 1353    seqcseq 10447
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-recs 6308  df-frec 6394  df-seqfrec 10448
This theorem is referenced by:  seqeq123d  10456  seq3f1olemstep  10503  seq3f1olemp  10504  exp3val  10524  sumeq1  11365  sumeq2  11369  summodc  11393  zsumdc  11394  fsum3  11397  isumz  11399  prodeq1f  11562  prodeq2w  11566  prodeq2  11567  prodmodc  11588  zproddc  11589  fprodseq  11593  prod1dc  11596  mulgval  12991  lgsval  14490  lgsval4  14506  lgsneg  14510  lgsmod  14512
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