Theorem intmin4 3617

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 3606	. . . 4 ⊢ (A ⊆ ∩ {x ∣ φ} ↔ ∀x(φ → A ⊆ x))
2		simpr 103	. . . . . . . 8 ⊢ ((A ⊆ x ∧ φ) → φ)
3		ancr 304	. . . . . . . 8 ⊢ ((φ → A ⊆ x) → (φ → (A ⊆ x ∧ φ)))
4	2, 3	impbid2 131	. . . . . . 7 ⊢ ((φ → A ⊆ x) → ((A ⊆ x ∧ φ) ↔ φ))
5	4	imbi1d 220	. . . . . 6 ⊢ ((φ → A ⊆ x) → (((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x)))
6	5	alimi 1324	. . . . 5 ⊢ (∀x(φ → A ⊆ x) → ∀x(((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x)))
7		albi 1337	. . . . 5 ⊢ (∀x(((A ⊆ x ∧ φ) → y ∈ x) ↔ (φ → y ∈ x)) → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
8	6, 7	syl 14	. . . 4 ⊢ (∀x(φ → A ⊆ x) → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
9	1, 8	sylbi 114	. . 3 ⊢ (A ⊆ ∩ {x ∣ φ} → (∀x((A ⊆ x ∧ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
10		vex 2538	. . . 4 ⊢ y ∈ V
11	10	elintab 3600	. . 3 ⊢ (y ∈ ∩ {x ∣ (A ⊆ x ∧ φ)} ↔ ∀x((A ⊆ x ∧ φ) → y ∈ x))
12	10	elintab 3600	. . 3 ⊢ (y ∈ ∩ {x ∣ φ} ↔ ∀x(φ → y ∈ x))
13	9, 11, 12	3bitr4g 212	. 2 ⊢ (A ⊆ ∩ {x ∣ φ} → (y ∈ ∩ {x ∣ (A ⊆ x ∧ φ)} ↔ y ∈ ∩ {x ∣ φ}))
14	13	eqrdv 2020	1 ⊢ (A ⊆ ∩ {x ∣ φ} → ∩ {x ∣ (A ⊆ x ∧ φ)} = ∩ {x ∣ φ})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374  {cab 2008   ⊆ wss 2894  ∩ cint 3589

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-int 3590

This theorem is referenced by: (None)