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Theorem elrint2 4997
Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
elrint2 (𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))
Distinct variable groups:   𝑦,𝐵   𝑦,𝑋
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem elrint2
StepHypRef Expression
1 elrint 4996 . 2 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
21baib 534 1 (𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  wral 3057  cin 3946   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-v 3473  df-in 3954  df-int 4952
This theorem is referenced by:  mreacs  17643
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