Theorem seqeq1d 14014

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 14011	. 2 ⊢ (𝐴 = 𝐵 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
3	1, 2	syl 17	1 ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

Syntax hints:   → wi 4   = wceq 1559  seqcseq 14008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-xp 5649  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-iota 6472  df-fv 6524  df-ov 7394  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-seq 14009

This theorem is referenced by:  seqeq123d  14017  seqf1olem2  14049  bcval5  14325  bcn2  14326  seqshft  15092  iserex  15675  isershft  15682  isercoll2  15687  isumsplit  15861  cvgrat  15904  ntrivcvg  15918  ntrivcvgtail  15921  fprodser  15970  eftlub  16132  gsumval2a  18710  gsumsgrpccat  18865  mulgnndir  19136  geolim3  26391  fmul01lt1lem2  46122  stirlinglem7  46615  stirlinglem12  46620