Theorem nfoprab2 7337

Description: The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)

Assertion

Ref	Expression
nfoprab2	⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Proof of Theorem nfoprab2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 7279	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2		nfe1 2147	. . . 4 ⊢ Ⅎ𝑦∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
3	2	nfex 2318	. . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
4	3	nfab 2913	. 2 ⊢ Ⅎ𝑦{𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
5	1, 4	nfcxfr 2905	1 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 396   = wceq 1539  ∃wex 1782  {cab 2715  Ⅎwnfc 2887  ⟨cop 4567  {coprab 7276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-oprab 7279

This theorem is referenced by:  ssoprab2b  7344  eqoprab2bw  7345  nfmpo2  7356  ov3  7435  tposoprab  8078