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Theorem seqeq123d 13366
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
Hypotheses
Ref Expression
seqeq123d.1 (𝜑𝑀 = 𝑁)
seqeq123d.2 (𝜑+ = 𝑄)
seqeq123d.3 (𝜑𝐹 = 𝐺)
Assertion
Ref Expression
seqeq123d (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))

Proof of Theorem seqeq123d
StepHypRef Expression
1 seqeq123d.1 . . 3 (𝜑𝑀 = 𝑁)
21seqeq1d 13363 . 2 (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
3 seqeq123d.2 . . 3 (𝜑+ = 𝑄)
43seqeq2d 13364 . 2 (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹))
5 seqeq123d.3 . . 3 (𝜑𝐹 = 𝐺)
65seqeq3d 13365 . 2 (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺))
72, 4, 63eqtrd 2857 1 (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  seqcseq 13357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seq 13358
This theorem is referenced by:  relexpsucnnr  14372  sseqval  31545  bj-finsumval0  34455
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