Theorem nfoprab 5863

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 5818	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2		nfv 1505	. . . . . . 7 ⊢ Ⅎ𝑤 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
3		nfoprab.1	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
4	2, 3	nfan 1542	. . . . . 6 ⊢ Ⅎ𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
5	4	nfex 1614	. . . . 5 ⊢ Ⅎ𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
6	5	nfex 1614	. . . 4 ⊢ Ⅎ𝑤∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
7	6	nfex 1614	. . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
8	7	nfab 2301	. 2 ⊢ Ⅎ𝑤{𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
9	1, 8	nfcxfr 2293	1 ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 103   = wceq 1332  Ⅎwnf 1437  ∃wex 1469  {cab 2140  Ⅎwnfc 2283  ⟨cop 3559  {coprab 5815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-oprab 5818

This theorem is referenced by:  nfmpo  5880