Theorem seqeq1d 14039

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 14036	. 2 ⊢ (𝐴 = 𝐵 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
3	1, 2	syl 18	1 ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

Syntax hints:   → wi 4   = wceq 1567  seqcseq 14033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5665  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-iota 6489  df-fv 6541  df-ov 7411  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-seq 14034

This theorem is referenced by:  seqeq123d  14042  seqf1olem2  14074  bcval5  14350  bcn2  14351  seqshft  15118  iserex  15704  isershft  15711  isercoll2  15716  isumsplit  15890  cvgrat  15933  ntrivcvg  15947  ntrivcvgtail  15950  fprodser  15999  eftlub  16161  gsumval2a  18739  gsumsgrpccat  18895  mulgnndir  19165  geolim3  26465  fmul01lt1lem2  46186  stirlinglem7  46679  stirlinglem12  46684