Theorem seqeq1d 13948

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 13945	. 2 ⊢ (𝐴 = 𝐵 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
3	1, 2	syl 17	1 ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

Syntax hints:   → wi 4   = wceq 1540  seqcseq 13942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-iota 6452  df-fv 6507  df-ov 7372  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-seq 13943

This theorem is referenced by:  seqeq123d  13951  seqf1olem2  13983  bcval5  14259  bcn2  14260  seqshft  15027  iserex  15599  isershft  15606  isercoll2  15611  isumsplit  15782  cvgrat  15825  ntrivcvg  15839  ntrivcvgtail  15842  fprodser  15891  eftlub  16053  gsumval2a  18588  gsumsgrpccat  18743  mulgnndir  19011  geolim3  26223  fmul01lt1lem2  45556  stirlinglem7  46051  stirlinglem12  46056