Theorem nfopab 5136

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 5131	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧 𝑤 = ⟨𝑥, 𝑦⟩
3		nfopab.1	. . . . . 6 ⊢ Ⅎ𝑧𝜑
4	2, 3	nfan 1900	. . . . 5 ⊢ Ⅎ𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5	4	nfex 2343	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6	5	nfex 2343	. . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7	6	nfab 2986	. 2 ⊢ Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
8	1, 7	nfcxfr 2977	1 ⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 398   = wceq 1537  ∃wex 1780  Ⅎwnf 1784  {cab 2801  Ⅎwnfc 2963  ⟨cop 4575  {copab 5130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-opab 5131

This theorem is referenced by:  nfmpt  5165  csbopab  5444  csbopabgALT  5445  nfxp  5590  nfco  5738  nfcnv  5751  nfofr  7417