Theorem hbopab2 2113

Description: The abstraction variables in a class abstraction of pairs are not free.

Assertion

Ref	Expression
hbopab2	⊢ (z ∈ {⟨x, y⟩∣φ} → ∀y z ∈ {⟨x, y⟩∣φ})

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

Proof of Theorem hbopab2

Step	Hyp	Ref	Expression
1		hbe1 709	. . . 4 ⊢ (∃y(w = ⟨x, y⟩ ∧ φ) → ∀y∃y(w = ⟨x, y⟩ ∧ φ))
2	1	hbex 701	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) → ∀y∃x∃y(w = ⟨x, y⟩ ∧ φ))
3	2	hbab 1096	. 2 ⊢ (z ∈ {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} → ∀y z ∈ {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)})
4		df-opab 2098	. . 3 ⊢ {⟨x, y⟩∣φ} = {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
5	4	eleq2i 1153	. 2 ⊢ (z ∈ {⟨x, y⟩∣φ} ↔ z ∈ {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)})
6	5	bial 695	. 2 ⊢ (∀y z ∈ {⟨x, y⟩∣φ} ↔ ∀y z ∈ {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)})
7	3, 5, 6	3imtr4 192	1 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀y z ∈ {⟨x, y⟩∣φ})

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab 2055

This theorem is referenced by:  opabsb 2114  opelopabg 2115  ssopab2 2119  dmopab 2539  rnopab 2566  cnvopab 2632  funopab 2694  zfrep6 2744  fnopabg 2745  fvopab2 2878  fopab2 2891  abrexexlem2 2911  dom2d 3307  xpmapenlem1 3391  xpmapenlem4 3394

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-opab 2098

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