Theorem hboprab2 3024

Description: The abstraction variables in an operation abstraction are not free.

Assertion

Ref	Expression
hboprab2	⊢ (w ∈ {⟨⟨x, y⟩, z⟩∣φ} → ∀y w ∈ {⟨⟨x, y⟩, z⟩∣φ})

Distinct variable group(s):   x,y,z   y,w

Proof of Theorem hboprab2

Step	Hyp	Ref	Expression
1		hbe1 709	. . . 4 ⊢ (∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀y∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
2	1	hbex 701	. . 3 ⊢ (∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀y∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
3	2	hbab 1096	. 2 ⊢ (w ∈ {v∣∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)} → ∀y w ∈ {v∣∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)})
4		df-oprab 3004	. . 3 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {v∣∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)}
5	4	eleq2i 1153	. 2 ⊢ (w ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ w ∈ {v∣∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)})
6	5	bial 695	. 2 ⊢ (∀y w ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ ∀y w ∈ {v∣∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)})
7	3, 5, 6	3imtr4 192	1 ⊢ (w ∈ {⟨⟨x, y⟩, z⟩∣φ} → ∀y w ∈ {⟨⟨x, y⟩, z⟩∣φ})

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab2 3002

This theorem is referenced by:  elrnoprab 3054  mapxpen 3390

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-oprab 3004

metamath.org