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Theorem elriin 3769
 Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
elriin (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝐵
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem elriin
StepHypRef Expression
1 elin 3166 . 2 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴𝐵 𝑥𝑋 𝑆))
2 eliin 3704 . . 3 (𝐵𝐴 → (𝐵 𝑥𝑋 𝑆 ↔ ∀𝑥𝑋 𝐵𝑆))
32pm5.32i 442 . 2 ((𝐵𝐴𝐵 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
41, 3bitri 182 1 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
 Colors of variables: wff set class Syntax hints:   ∧ wa 102   ↔ wb 103   ∈ wcel 1434  ∀wral 2353   ∩ cin 2982  ∩ ciin 3700 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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2613  df-in 2989  df-iin 3702 This theorem is referenced by: (None)
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