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Theorem seqeq1d 12069
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
seqeq1d  |-  ( ph  ->  seq A (  .+  ,  F )  =  seq B (  .+  ,  F ) )

Proof of Theorem seqeq1d
StepHypRef Expression
1 seqeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 seqeq1 12066 . 2  |-  ( A  =  B  ->  seq A (  .+  ,  F )  =  seq B (  .+  ,  F ) )
31, 2syl 16 1  |-  ( ph  ->  seq A (  .+  ,  F )  =  seq B (  .+  ,  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    seqcseq 12063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fv 5587  df-recs 7032  df-rdg 7066  df-seq 12064
This theorem is referenced by:  seqeq123d  12072  seqf1olem2  12103  bcval5  12351  bcn2  12352  seqshft  12868  iserex  13428  isershft  13435  isercoll2  13440  isumsplit  13604  cvgrat  13644  eftlub  13694  gsumval2a  15818  gsumccat  15825  mulgnndir  15957  geolim3  22462  ntrivcvg  28458  ntrivcvgtail  28461  fprodser  28508  fmul01lt1lem2  30954  stirlinglem7  31199  stirlinglem12  31204
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