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Theorem elrint2 3706
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 3705 . 2 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
21baib 864 1 (𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1436  wral 2355  cin 2985   cint 3665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-in 2992  df-int 3666
This theorem is referenced by: (None)
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