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Theorem seqeq3d 10452
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 10449 . 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 10444
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 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-recs 6305  df-frec 6391  df-seqfrec 10445
This theorem is referenced by:  seqeq123d  10453  seq3f1olemstep  10500  seq3f1olemp  10501  exp3val  10521  sumeq1  11362  sumeq2  11366  summodc  11390  zsumdc  11391  fsum3  11394  isumz  11396  prodeq1f  11559  prodeq2w  11563  prodeq2  11564  prodmodc  11585  zproddc  11586  fprodseq  11590  prod1dc  11593  mulgval  12985  lgsval  14375  lgsval4  14391  lgsneg  14395  lgsmod  14397
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