Theorem hbrab 1311

Description: A variable not free in a wff remains so in a restricted class abstraction.

Hypotheses

Ref	Expression
hbrab.1	⊢ (φ → ∀xφ)
hbrab.2	⊢ (z ∈ A → ∀x z ∈ A)

Assertion

Ref	Expression
hbrab	⊢ (z ∈ {y ∈ A∣φ} → ∀x z ∈ {y ∈ A∣φ})

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

Proof of Theorem hbrab

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (z ∈ y → ∀x z ∈ y)
2		hbrab.2	. . . . 5 ⊢ (z ∈ A → ∀x z ∈ A)
3	1, 2	hbel 1172	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
4		hbrab.1	. . . 4 ⊢ (φ → ∀xφ)
5	3, 4	hban 704	. . 3 ⊢ ((y ∈ A ∧ φ) → ∀x(y ∈ A ∧ φ))
6	5	hbab 1096	. 2 ⊢ (z ∈ {y∣(y ∈ A ∧ φ)} → ∀x z ∈ {y∣(y ∈ A ∧ φ)})
7		df-rab 1208	. . 3 ⊢ {y ∈ A∣φ} = {y∣(y ∈ A ∧ φ)}
8	7	eleq2i 1153	. 2 ⊢ (z ∈ {y ∈ A∣φ} ↔ z ∈ {y∣(y ∈ A ∧ φ)})
9	8	bial 695	. 2 ⊢ (∀x z ∈ {y ∈ A∣φ} ↔ ∀x z ∈ {y∣(y ∈ A ∧ φ)})
10	6, 8, 9	3imtr4 192	1 ⊢ (z ∈ {y ∈ A∣φ} → ∀x z ∈ {y ∈ A∣φ})

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   ∈ wel 803  {cab 1090   ∈ wcel 1092  {crab 1204

This theorem is referenced by:  scottex 3541

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-rab 1208

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