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Theorem seqeq1d 11804
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 11801 . 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 1369    seqcseq 11798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-cnv 4843  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fv 5421  df-recs 6824  df-rdg 6858  df-seq 11799
This theorem is referenced by:  seqeq123d  11807  seqf1olem2  11838  bcval5  12086  bcn2  12087  seqshft  12566  iserex  13126  isershft  13133  isercoll2  13138  isumsplit  13295  cvgrat  13335  eftlub  13385  gsumval2a  15503  gsumccat  15510  mulgnndir  15640  geolim3  21785  ntrivcvg  27381  ntrivcvgtail  27384  fprodser  27431  fmul01lt1lem2  29737  stirlinglem7  29846  stirlinglem12  29851
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