Theorem seqeq1d 14150

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 14147	. 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 14144

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 2733

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  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 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-iota 6494  df-fv 6546  df-ov 7423  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-seq 14145

This theorem is used by:  seqeq123d  14153  seqf1olem2  14185  bcval5  14462  bcn2  14463  seqshft  15238  iserex  15824  isershft  15831  isercoll2  15836  isumsplit  16009  cvgrat  16052  ntrivcvg  16066  ntrivcvgtail  16069  fprodser  16116  eftlub  16277  gsumval2a  18874  gsumsgrpccat  19036  mulgnndir  19313  geolim3  26666  fmul01lt1lem2  46596  stirlinglem7  47089  stirlinglem12  47094