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Theorem seqeq1d 11913
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 11910 . 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 1370    seqcseq 11907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-cnv 4946  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fv 5524  df-recs 6932  df-rdg 6966  df-seq 11908
This theorem is referenced by:  seqeq123d  11916  seqf1olem2  11947  bcval5  12195  bcn2  12196  seqshft  12676  iserex  13236  isershft  13243  isercoll2  13248  isumsplit  13405  cvgrat  13445  eftlub  13495  gsumval2a  15614  gsumccat  15621  mulgnndir  15751  geolim3  21921  ntrivcvg  27546  ntrivcvgtail  27549  fprodser  27596  fmul01lt1lem2  29904  stirlinglem7  30013  stirlinglem12  30018
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