Theorem nfopab 4628

Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed 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 4624	. 2 ⊢ {⟨x, y⟩ ∣ φ} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
2		nfv 1619	. . . . . 6 ⊢ Ⅎz w = ⟨x, y⟩
3		nfopab.1	. . . . . 6 ⊢ Ⅎzφ
4	2, 3	nfan 1824	. . . . 5 ⊢ Ⅎz(w = ⟨x, y⟩ ∧ φ)
5	4	nfex 1843	. . . 4 ⊢ Ⅎz∃y(w = ⟨x, y⟩ ∧ φ)
6	5	nfex 1843	. . 3 ⊢ Ⅎz∃x∃y(w = ⟨x, y⟩ ∧ φ)
7	6	nfab 2494	. 2 ⊢ Ⅎz{w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
8	1, 7	nfcxfr 2487	1 ⊢ Ⅎz{⟨x, y⟩ ∣ φ}

Syntax hints:   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  Ⅎwnfc 2477  ⟨cop 4562  {copab 4623

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-opab 4624

This theorem is referenced by:  csbopabg  4638  nfxp  4811  nfco  4883  nfcnv  4892  nfmpt  5672