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Theorem seqeq1d 12747
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
Hypothesis
Ref Expression
seqeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
seqeq1d (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

Proof of Theorem seqeq1d
StepHypRef Expression
1 seqeqd.1 . 2 (𝜑𝐴 = 𝐵)
2 seqeq1 12744 . 2 (𝐴 = 𝐵 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
31, 2syl 17 1 (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  seqcseq 12741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fv 5855  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seq 12742
This theorem is referenced by:  seqeq123d  12750  seqf1olem2  12781  bcval5  13045  bcn2  13046  seqshft  13759  iserex  14321  isershft  14328  isercoll2  14333  isumsplit  14497  cvgrat  14540  ntrivcvg  14554  ntrivcvgtail  14557  fprodser  14604  eftlub  14764  gsumval2a  17200  gsumccat  17299  mulgnndir  17490  mulgnndirOLD  17491  geolim3  23998  fmul01lt1lem2  39221  stirlinglem7  39604  stirlinglem12  39609
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