Theorem intmin4 4867

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 4855	. . . 4 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥))
2		simpr 488	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝜑)
3		ancr 550	. . . . . . . 8 ⊢ ((𝜑 → 𝐴 ⊆ 𝑥) → (𝜑 → (𝐴 ⊆ 𝑥 ∧ 𝜑)))
4	2, 3	impbid2 229	. . . . . . 7 ⊢ ((𝜑 → 𝐴 ⊆ 𝑥) → ((𝐴 ⊆ 𝑥 ∧ 𝜑) ↔ 𝜑))
5	4	imbi1d 345	. . . . . 6 ⊢ ((𝜑 → 𝐴 ⊆ 𝑥) → (((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝑦 ∈ 𝑥)))
6	5	alimi 1813	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝐴 ⊆ 𝑥) → ∀𝑥(((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝑦 ∈ 𝑥)))
7		albi 1820	. . . . 5 ⊢ (∀𝑥(((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝑦 ∈ 𝑥)) → (∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝜑 → 𝑦 ∈ 𝑥)))
8	6, 7	syl 17	. . . 4 ⊢ (∀𝑥(𝜑 → 𝐴 ⊆ 𝑥) → (∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝜑 → 𝑦 ∈ 𝑥)))
9	1, 8	sylbi 220	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} → (∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝜑 → 𝑦 ∈ 𝑥)))
10		vex 3413	. . . 4 ⊢ 𝑦 ∈ V
11	10	elintab 4849	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝑦 ∈ 𝑥))
12	10	elintab 4849	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝑦 ∈ 𝑥))
13	9, 11, 12	3bitr4g 317	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} → (𝑦 ∈ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ 𝑦 ∈ ∩ {𝑥 ∣ 𝜑}))
14	13	eqrdv 2756	1 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735   ⊆ wss 3858  ∩ cint 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-in 3865  df-ss 3875  df-int 4839

This theorem is referenced by: (None)