Theorem elintrabg 3940

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)

Assertion

Ref	Expression
elintrabg	⊢ (A ∈ V → (A ∈ ∩{x ∈ B ∣ φ} ↔ ∀x ∈ B (φ → A ∈ x)))

Distinct variable group:   x,A

Allowed substitution hints:   φ(x)   B(x)   V(x)

Proof of Theorem elintrabg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. 2 ⊢ (y = A → (y ∈ ∩{x ∈ B ∣ φ} ↔ A ∈ ∩{x ∈ B ∣ φ}))
2		eleq1 2413	. . . 4 ⊢ (y = A → (y ∈ x ↔ A ∈ x))
3	2	imbi2d 307	. . 3 ⊢ (y = A → ((φ → y ∈ x) ↔ (φ → A ∈ x)))
4	3	ralbidv 2635	. 2 ⊢ (y = A → (∀x ∈ B (φ → y ∈ x) ↔ ∀x ∈ B (φ → A ∈ x)))
5		vex 2863	. . 3 ⊢ y ∈ V
6	5	elintrab 3939	. 2 ⊢ (y ∈ ∩{x ∈ B ∣ φ} ↔ ∀x ∈ B (φ → y ∈ x))
7	1, 4, 6	vtoclbg 2916	1 ⊢ (A ∈ V → (A ∈ ∩{x ∈ B ∣ φ} ↔ ∀x ∈ B (φ → A ∈ x)))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2615  {crab 2619  ∩cint 3927

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 2479  df-ral 2620  df-rab 2624  df-v 2862  df-int 3928

This theorem is referenced by: (None)