Theorem elrint2 4997

Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
elrint2	⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))

Distinct variable groups:   𝑦,𝐵   𝑦,𝑋

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem elrint2

Step	Hyp	Ref	Expression
1		elrint 4996	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
2	1	baib 537	1 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2107  ∀wral 3062   ∩ cin 3948  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3956  df-int 4952

This theorem is referenced by:  mreacs  17602