Theorem intmin4 3956

Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)

Assertion

Ref	Expression
intmin4	⊢ (A ⊆ ∩{x ∣ φ} → ∩{x ∣ (A ⊆ x ∧ φ)} = ∩{x ∣ φ})

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem intmin4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssintab 3944	. . . 4 ⊢ (A ⊆ ∩{x ∣ φ} ↔ ∀x(φ → A ⊆ x))
2		simpr 447	. . . . . . . 8 ⊢ ((A ⊆ x ∧ φ) → φ)
3		ancr 532	. . . . . . . 8 ⊢ ((φ → A ⊆ x) → (φ → (A ⊆ x ∧ φ)))
4	2, 3	impbid2 195	. . . . . . 7 ⊢ ((φ → A ⊆ x) → ((A ⊆ x ∧ φ) ↔ φ))
5	4	imbi1d 308	. . . . . 6 ⊢ ((φ → A ⊆ x) → (((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x)))
6	5	alimi 1559	. . . . 5 ⊢ (∀x(φ → A ⊆ x) → ∀x(((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x)))
7		albi 1564	. . . . 5 ⊢ (∀x(((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x)) → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
8	6, 7	syl 15	. . . 4 ⊢ (∀x(φ → A ⊆ x) → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
9	1, 8	sylbi 187	. . 3 ⊢ (A ⊆ ∩{x ∣ φ} → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
10		vex 2863	. . . 4 ⊢ y ∈ V
11	10	elintab 3938	. . 3 ⊢ (y ∈ ∩{x ∣ (A ⊆ x ∧ φ)} ↔ ∀x((A ⊆ x ∧ φ) → y ∈ x))
12	10	elintab 3938	. . 3 ⊢ (y ∈ ∩{x ∣ φ} ↔ ∀x(φ → y ∈ x))
13	9, 11, 12	3bitr4g 279	. 2 ⊢ (A ⊆ ∩{x ∣ φ} → (y ∈ ∩{x ∣ (A ⊆ x ∧ φ)} ↔ y ∈ ∩{x ∣ φ}))
14	13	eqrdv 2351	1 ⊢ (A ⊆ ∩{x ∣ φ} → ∩{x ∣ (A ⊆ x ∧ φ)} = ∩{x ∣ φ})

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339   ⊆ wss 3258  ∩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-nan 1288  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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by: (None)