Theorem iinab 4027

Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)

Assertion

Ref	Expression
iinab	⊢ ∩x ∈ A {y ∣ φ} = {y ∣ ∀x ∈ A φ}

Distinct variable groups:   y,A   x,y

Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem iinab

Step	Hyp	Ref	Expression
1		nfcv 2489	. . . 4 ⊢ ℲyA
2		nfab1 2491	. . . 4 ⊢ Ⅎy{y ∣ φ}
3	1, 2	nfiin 3996	. . 3 ⊢ Ⅎy∩x ∈ A {y ∣ φ}
4		nfab1 2491	. . 3 ⊢ Ⅎy{y ∣ ∀x ∈ A φ}
5	3, 4	cleqf 2513	. 2 ⊢ (∩x ∈ A {y ∣ φ} = {y ∣ ∀x ∈ A φ} ↔ ∀y(y ∈ ∩x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∀x ∈ A φ}))
6		abid 2341	. . . 4 ⊢ (y ∈ {y ∣ φ} ↔ φ)
7	6	ralbii 2638	. . 3 ⊢ (∀x ∈ A y ∈ {y ∣ φ} ↔ ∀x ∈ A φ)
8		vex 2862	. . . 4 ⊢ y ∈ V
9		eliin 3974	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A {y ∣ φ} ↔ ∀x ∈ A y ∈ {y ∣ φ}))
10	8, 9	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ A {y ∣ φ} ↔ ∀x ∈ A y ∈ {y ∣ φ})
11		abid 2341	. . 3 ⊢ (y ∈ {y ∣ ∀x ∈ A φ} ↔ ∀x ∈ A φ)
12	7, 10, 11	3bitr4i 268	. 2 ⊢ (y ∈ ∩x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∀x ∈ A φ})
13	5, 12	mpgbir 1550	1 ⊢ ∩x ∈ A {y ∣ φ} = {y ∣ ∀x ∈ A φ}

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  Vcvv 2859  ∩ciin 3970

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-or 359  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-nfc 2478  df-ral 2619  df-v 2861  df-iin 3972

This theorem is referenced by:  iinrab  4028