Theorem nfoprab 7474

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 7414	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑤 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
3		nfoprab.1	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
4	2, 3	nfan 1929	. . . . . 6 ⊢ Ⅎ𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
5	4	nfex 2357	. . . . 5 ⊢ Ⅎ𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
6	5	nfex 2357	. . . 4 ⊢ Ⅎ𝑤∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
7	6	nfex 2357	. . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
8	7	nfab 2931	. 2 ⊢ Ⅎ𝑤{𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
9	1, 8	nfcxfr 2923	1 ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809  Ⅎwnf 1813  {cab 2741  Ⅎwnfc 2910  ⟨cop 4595  {coprab 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-oprab 7414

This theorem is referenced by:  nfmpo  7492