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Theorem seqeq1d 12157
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 12154 . 2  |-  ( A  =  B  ->  seq A (  .+  ,  F )  =  seq B (  .+  ,  F ) )
31, 2syl 17 1  |-  ( ph  ->  seq A (  .+  ,  F )  =  seq B (  .+  ,  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    seqcseq 12151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-xp 4829  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-iota 5533  df-fv 5577  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-seq 12152
This theorem is referenced by:  seqeq123d  12160  seqf1olem2  12191  bcval5  12440  bcn2  12441  seqshft  13067  iserex  13628  isershft  13635  isercoll2  13640  isumsplit  13803  cvgrat  13844  ntrivcvg  13858  ntrivcvgtail  13861  fprodser  13908  eftlub  14053  gsumval2a  16230  gsumccat  16333  mulgnndir  16488  geolim3  23027  fmul01lt1lem2  36947  stirlinglem7  37230  stirlinglem12  37235
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