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Theorem seqeq123d 14017
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 14014 . 2 (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
3 seqeq123d.2 . . 3 (𝜑+ = 𝑄)
43seqeq2d 14015 . 2 (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹))
5 seqeq123d.3 . . 3 (𝜑𝐹 = 𝐺)
65seqeq3d 14016 . 2 (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺))
72, 4, 63eqtrd 2772 1 (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  seqcseq 14008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-xp 5688  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-iota 6505  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-seq 14009
This theorem is referenced by:  relexpsucnnr  15014  sseqval  34049  bj-finsumval0  36805  itcoval  47830
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