Theorem elintdv 45827

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

Hypotheses

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

Assertion

Ref	Expression
elintdv	⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elintdv

Step	Hyp	Ref	Expression
1		nfv 1944	. 2 ⊢ Ⅎ𝑥𝜑
2		elintdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		elintdv.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)
4	1, 2, 3	elintd 45822	1 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2143  ∩ cint 4912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-int 4913

This theorem is used by: (None)