Theorem elintd 45822

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 417	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
4	1, 3	ralrimi 3262	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)
5		elintd.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		elintg 4919	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))
7	5, 6	syl 18	. 2 ⊢ (𝜑 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))
8	4, 7	mpbird 260	1 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  Ⅎwnf 1812   ∈ wcel 2142  ∀wral 3078  ∩ cint 4911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-int 4912

This theorem is used by:  ssuniint  45826  elintdv  45827