Theorem elintd 42624

Description: Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
elintd.1	⊢ Ⅎ𝑥𝜑
elintd.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
elintd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)

Assertion

Ref	Expression
elintd	⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem elintd

Step	Hyp	Ref	Expression
1		elintd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		elintd.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)
3	2	ex 413	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
4	1, 3	ralrimi 3141	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)
5		elintd.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		elintg 4887	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))
7	5, 6	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))
8	4, 7	mpbird 256	1 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  Ⅎwnf 1786   ∈ wcel 2106  ∀wral 3064  ∩ cint 4879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-int 4880

This theorem is referenced by:  ssuniint  42628  elintdv  42629