Theorem nfoprab 7497

Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)

Hypothesis

Ref	Expression
nfoprab.1	⊢ Ⅎ𝑤𝜑

Assertion

Ref	Expression
nfoprab	⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Distinct variable groups:   𝑥,𝑤   𝑦,𝑤   𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem nfoprab

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 7435	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑤 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
3		nfoprab.1	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
4	2, 3	nfan 1897	. . . . . 6 ⊢ Ⅎ𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
5	4	nfex 2323	. . . . 5 ⊢ Ⅎ𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
6	5	nfex 2323	. . . 4 ⊢ Ⅎ𝑤∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
7	6	nfex 2323	. . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
8	7	nfab 2909	. 2 ⊢ Ⅎ𝑤{𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
9	1, 8	nfcxfr 2901	1 ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1776  Ⅎwnf 1780  {cab 2712  Ⅎwnfc 2888  ⟨cop 4637  {coprab 7432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-oprab 7435

This theorem is referenced by:  nfmpo  7515