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Theorem nfmpt2 6721
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 6652 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
2 nfmpt2.1 . . . . . 6 𝑧𝐴
32nfcri 2757 . . . . 5 𝑧 𝑥𝐴
4 nfmpt2.2 . . . . . 6 𝑧𝐵
54nfcri 2757 . . . . 5 𝑧 𝑦𝐵
63, 5nfan 1827 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
7 nfmpt2.3 . . . . 5 𝑧𝐶
87nfeq2 2779 . . . 4 𝑧 𝑤 = 𝐶
96, 8nfan 1827 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)
109nfoprab 6704 . 2 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
111, 10nfcxfr 2761 1 𝑧(𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1482  wcel 1989  wnfc 2750  {coprab 6648  cmpt2 6649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-oprab 6651  df-mpt2 6652
This theorem is referenced by:  el2mpt2csbcl  7247  nfseq  12806  ptbasfi  21378  numclwlk1lem2  27214  sdclem1  33519  fmuldfeqlem1  39620  stoweidlem51  40037  vonicc  40668
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