Theorem elintdv 45898

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 1947	. 2 ⊢ Ⅎ𝑥𝜑
2		elintdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		elintdv.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)
4	1, 2, 3	elintd 45893	1 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∩ cint 4910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-int 4911

This theorem is used by: (None)