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Theorem nfmpt2 6622
 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 6554 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
2 nfmpt2.1 . . . . . 6 𝑧𝐴
32nfcri 2745 . . . . 5 𝑧 𝑥𝐴
4 nfmpt2.2 . . . . . 6 𝑧𝐵
54nfcri 2745 . . . . 5 𝑧 𝑦𝐵
63, 5nfan 1816 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
7 nfmpt2.3 . . . . 5 𝑧𝐶
87nfeq2 2766 . . . 4 𝑧 𝑤 = 𝐶
96, 8nfan 1816 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)
109nfoprab 6605 . 2 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
111, 10nfcxfr 2749 1 𝑧(𝑥𝐴, 𝑦𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  {coprab 6550   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  el2mpt2csbcl  7137  nfseq  12673  ptbasfi  21194  sdclem1  32709  fmuldfeqlem1  38649  stoweidlem51  38944  vonicc  39576
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