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Theorem nfmpt2 5604
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
nfmpt2.1 𝑧𝐴
nfmpt2.2 𝑧𝐵
nfmpt2.3 𝑧𝐶
Assertion
Ref Expression
nfmpt2 𝑧(𝑥𝐴, 𝑦𝐵𝐶)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem nfmpt2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-mpt2 5548 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
2 nfmpt2.1 . . . . . 6 𝑧𝐴
32nfcri 2214 . . . . 5 𝑧 𝑥𝐴
4 nfmpt2.2 . . . . . 6 𝑧𝐵
54nfcri 2214 . . . . 5 𝑧 𝑦𝐵
63, 5nfan 1498 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
7 nfmpt2.3 . . . . 5 𝑧𝐶
87nfeq2 2231 . . . 4 𝑧 𝑤 = 𝐶
96, 8nfan 1498 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)
109nfoprab 5588 . 2 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
111, 10nfcxfr 2217 1 𝑧(𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wcel 1434  wnfc 2207  {coprab 5544  cmpt2 5545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-oprab 5547  df-mpt2 5548
This theorem is referenced by:  nfiseq  9528
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