Theorem seqeq1d 14190

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 14187	. 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 14184

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-xp 5664  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-iota 6520  df-fv 6574  df-ov 7454  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seq 14185

This theorem is used by:  seqeq123d  14193  seqf1olem2  14225  bcval5  14503  bcn2  14504  seqshft  15281  iserex  15868  isershft  15875  isercoll2  15880  isumsplit  16053  cvgrat  16096  ntrivcvg  16110  ntrivcvgtail  16113  fprodser  16160  eftlub  16321  gsumval2a  18924  gsumsgrpccat  19086  mulgnndir  19363  geolim3  26723  fmul01lt1lem2  46870  stirlinglem7  47363  stirlinglem12  47368