Theorem seqeq1d 14071

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 14068	. 2 ⊢ (𝐴 = 𝐵 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
3	1, 2	syl 18	1 ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  seqcseq 14065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5661  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-iota 6489  df-fv 6541  df-ov 7416  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-seq 14066

This theorem is used by:  seqeq123d  14074  seqf1olem2  14106  bcval5  14382  bcn2  14383  seqshft  15158  iserex  15744  isershft  15751  isercoll2  15756  isumsplit  15929  cvgrat  15972  ntrivcvg  15986  ntrivcvgtail  15989  fprodser  16036  eftlub  16197  gsumval2a  18787  gsumsgrpccat  18949  mulgnndir  19226  geolim3  26575  fmul01lt1lem2  46415  stirlinglem7  46908  stirlinglem12  46913