Theorem seqeq1d 14056

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 14053	. 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 14050

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 2148  ax-9 2156  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fv 6548  df-ov 7419  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-seq 14051

This theorem is used by:  seqeq123d  14059  seqf1olem2  14091  bcval5  14367  bcn2  14368  seqshft  15141  iserex  15727  isershft  15734  isercoll2  15739  isumsplit  15912  cvgrat  15955  ntrivcvg  15969  ntrivcvgtail  15972  fprodser  16021  eftlub  16182  gsumval2a  18764  gsumsgrpccat  18922  mulgnndir  19192  geolim3  26531  fmul01lt1lem2  46334  stirlinglem7  46827  stirlinglem12  46832