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Theorem seqeq123d 10473
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 10470 . 2 (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
3 seqeq123d.2 . . 3 (𝜑+ = 𝑄)
43seqeq2d 10471 . 2 (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹))
5 seqeq123d.3 . . 3 (𝜑𝐹 = 𝐺)
65seqeq3d 10472 . 2 (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺))
72, 4, 63eqtrd 2226 1 (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  seqcseq 10464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-cnv 4649  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-recs 6324  df-frec 6410  df-seqfrec 10465
This theorem is referenced by: (None)
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