Theorem hboprab 5563

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

Hypothesis

Ref	Expression
hboprab.1	⊢ (φ → ∀wφ)

Assertion

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

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

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

Proof of Theorem hboprab

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		ax-17 1616	. . . . . . 7 ⊢ (v = ⟨⟨x, y⟩, z⟩ → ∀w v = ⟨⟨x, y⟩, z⟩)
3		hboprab.1	. . . . . . 7 ⊢ (φ → ∀wφ)
4	2, 3	hban 1828	. . . . . 6 ⊢ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀w(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
5	4	hbex 1841	. . . . 5 ⊢ (∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀w∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
6	5	hbex 1841	. . . 4 ⊢ (∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀w∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
7	6	hbex 1841	. . 3 ⊢ (∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀w∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
8	7	hbab 2344	. 2 ⊢ (u ∈ {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)} → ∀w u ∈ {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)})
9	1, 8	hbxfreq 2457	1 ⊢ (u ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} → ∀w u ∈ {⟨⟨x, y⟩, z⟩ ∣ φ})

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ⟨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-oprab 5529

This theorem is used by: (None)