Theorem nfoprab 5550

Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)

Hypothesis

Ref	Expression
nfoprab.1	⊢ Ⅎwφ

Assertion

Ref	Expression
nfoprab	⊢ Ⅎw{⟨⟨x, y⟩, z⟩ ∣ φ}

Distinct variable groups:   x,w   y,w   z,w

Allowed substitution hints:   φ(x, y, z, w)

Proof of Theorem nfoprab

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 5529	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)}
2		nfv 1619	. . . . . . 7 ⊢ Ⅎw v = ⟨⟨x, y⟩, z⟩
3		nfoprab.1	. . . . . . 7 ⊢ Ⅎwφ
4	2, 3	nfan 1824	. . . . . 6 ⊢ Ⅎw(v = ⟨⟨x, y⟩, z⟩ ∧ φ)
5	4	nfex 1843	. . . . 5 ⊢ Ⅎw∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)
6	5	nfex 1843	. . . 4 ⊢ Ⅎw∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)
7	6	nfex 1843	. . 3 ⊢ Ⅎw∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)
8	7	nfab 2494	. 2 ⊢ Ⅎw{v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)}
9	1, 8	nfcxfr 2487	1 ⊢ Ⅎw{⟨⟨x, y⟩, z⟩ ∣ φ}

This proof depends on syntax axioms:   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  Ⅎwnfc 2477  ⟨cop 4562  {coprab 5528

This proof depends on 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 proof 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 used by:  nfmpt2  5676