Theorem hboprab3 5562

Description: The abstraction variables in an operation class abstraction are not free. (Contributed by set.mm contributors, 22-Aug-2013.)

Assertion

Ref	Expression
hboprab3	⊢ (w ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} → ∀z w ∈ {⟨⟨x, y⟩, z⟩ ∣ φ})

Distinct variable group:   z,w

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

Proof of Theorem hboprab3

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		hbe1 1731	. . . . 5 ⊢ (∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀z∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
3	2	hbex 1841	. . . 4 ⊢ (∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀z∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
4	3	hbex 1841	. . 3 ⊢ (∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀z∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
5	4	hbab 2344	. 2 ⊢ (w ∈ {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)} → ∀z w ∈ {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)})
6	1, 5	hbxfreq 2457	1 ⊢ (w ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} → ∀z w ∈ {⟨⟨x, y⟩, z⟩ ∣ φ})

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ⟨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-oprab 5529

This theorem is referenced by: (None)