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Theorem seqeq123d 10467
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 10464 . 2 (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
3 seqeq123d.2 . . 3 (𝜑+ = 𝑄)
43seqeq2d 10465 . 2 (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹))
5 seqeq123d.3 . . 3 (𝜑𝐹 = 𝐺)
65seqeq3d 10466 . 2 (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺))
72, 4, 63eqtrd 2224 1 (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  seqcseq 10458
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-recs 6319  df-frec 6405  df-seqfrec 10459
This theorem is referenced by: (None)
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