Theorem nfopab 4157

Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)

Hypothesis

Ref	Expression
nfopab.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfopab	⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem nfopab

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 4151	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfv 1576	. . . . . 6 ⊢ Ⅎ𝑧 𝑤 = ⟨𝑥, 𝑦⟩
3		nfopab.1	. . . . . 6 ⊢ Ⅎ𝑧𝜑
4	2, 3	nfan 1613	. . . . 5 ⊢ Ⅎ𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5	4	nfex 1685	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6	5	nfex 1685	. . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7	6	nfab 2379	. 2 ⊢ Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
8	1, 7	nfcxfr 2371	1 ⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 104   = wceq 1397  Ⅎwnf 1508  ∃wex 1540  {cab 2217  Ⅎwnfc 2361  ⟨cop 3672  {copab 4149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-opab 4151

This theorem is referenced by:  csbopabg  4167  nfmpt  4181  nfxp  4752  nfco  4895  nfcnv  4909  nfofr  6241