Theorem nfoprab1 7459

Description: The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Assertion

Ref	Expression
nfoprab1	⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Proof of Theorem nfoprab1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 7402	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2		nfe1 2186	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
3	2	nfab 2932	. 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
4	1, 3	nfcxfr 2924	1 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 399   = wceq 1562  ∃wex 1801  {cab 2742  Ⅎwnfc 2911  ⟨cop 4590  {coprab 7399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-oprab 7402

This theorem is referenced by:  ssoprab2b  7467  eqoprab2bw  7468  nfmpo1  7478  ov3  7561  tposoprab  8244