Theorem nfoprab3 7431

Description: The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)

Assertion

Ref	Expression
nfoprab3	⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Proof of Theorem nfoprab3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 7372	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2		nfe1 2156	. . . . 5 ⊢ Ⅎ𝑧∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
3	2	nfex 2330	. . . 4 ⊢ Ⅎ𝑧∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
4	3	nfex 2330	. . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
5	4	nfab 2905	. 2 ⊢ Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
6	1, 5	nfcxfr 2897	1 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  Ⅎwnfc 2884  ⟨cop 4588  {coprab 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-oprab 7372

This theorem is referenced by:  ssoprab2b  7437  eqoprab2bw  7438  ov3  7531  tposoprab  8214