Theorem seqeq1d 14048

Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Hypothesis

Ref	Expression
seqeqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
seqeq1d	⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

Proof of Theorem seqeq1d

Step	Hyp	Ref	Expression
1		seqeqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		seqeq1 14045	. 2 ⊢ (𝐴 = 𝐵 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
3	1, 2	syl 17	1 ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

Syntax hints:   → wi 4   = wceq 1540  seqcseq 14042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seq 14043

This theorem is referenced by:  seqeq123d  14051  seqf1olem2  14083  bcval5  14357  bcn2  14358  seqshft  15124  iserex  15693  isershft  15700  isercoll2  15705  isumsplit  15876  cvgrat  15919  ntrivcvg  15933  ntrivcvgtail  15936  fprodser  15985  eftlub  16145  gsumval2a  18698  gsumsgrpccat  18853  mulgnndir  19121  geolim3  26381  fmul01lt1lem2  45600  stirlinglem7  46095  stirlinglem12  46100