Theorem elintdv 42518

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 1918	. 2 ⊢ Ⅎ𝑥𝜑
2		elintdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		elintdv.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)
4	1, 2, 3	elintd 42513	1 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-int 4877

This theorem is referenced by: (None)