Theorem elintd 45119

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 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
4	1, 3	ralrimi 3230	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)
5		elintd.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		elintg 4903	. . 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 257	1 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2111  ∀wral 3047  ∩ cint 4895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-int 4896

This theorem is referenced by:  ssuniint  45123  elintdv  45124