Theorem hbopab 2111

Description: Bound-variable hypothesis builder for class abstraction.

Hypothesis

Ref	Expression
hbopab.1	⊢ (φ → ∀zφ)

Assertion

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

Distinct variable group(s):   x,y,z,w

Proof of Theorem hbopab

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (w = ⟨x, y⟩ → ∀z w = ⟨x, y⟩)
2		hbopab.1	. . . . 5 ⊢ (φ → ∀zφ)
3	1, 2	hban 704	. . . 4 ⊢ ((w = ⟨x, y⟩ ∧ φ) → ∀z(w = ⟨x, y⟩ ∧ φ))
4	3	hbex 701	. . 3 ⊢ (∃y(w = ⟨x, y⟩ ∧ φ) → ∀z∃y(w = ⟨x, y⟩ ∧ φ))
5	4	hbex 701	. 2 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) → ∀z∃x∃y(w = ⟨x, y⟩ ∧ φ))
6		elopab 2110	. 2 ⊢ (w ∈ {⟨x, y⟩∣φ} ↔ ∃x∃y(w = ⟨x, y⟩ ∧ φ))
7	6	bial 695	. 2 ⊢ (∀z w ∈ {⟨x, y⟩∣φ} ↔ ∀z∃x∃y(w = ⟨x, y⟩ ∧ φ))
8	5, 6, 7	3imtr4 192	1 ⊢ (w ∈ {⟨x, y⟩∣φ} → ∀z w ∈ {⟨x, y⟩∣φ})

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

This theorem is referenced by:  hbco 2508  hbrdg 2974  mapxpen 3390  tz9.12lem3 3505

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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