Theorem nfoprab2 5548

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 5529	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)}
2		nfe1 1732	. . . 4 ⊢ Ⅎy∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)
3	2	nfex 1843	. . 3 ⊢ Ⅎy∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)
4	3	nfab 2494	. 2 ⊢ Ⅎy{w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)}
5	1, 4	nfcxfr 2487	1 ⊢ Ⅎy{⟨⟨x, y⟩, z⟩ ∣ φ}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  {cab 2339  Ⅎwnfc 2477  ⟨cop 4562  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-oprab 5529

This theorem is referenced by:  ov3  5600  nfmpt22  5675