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Theorem elriin 3972
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 3333 . 2 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴𝐵 𝑥𝑋 𝑆))
2 eliin 3906 . . 3 (𝐵𝐴 → (𝐵 𝑥𝑋 𝑆 ↔ ∀𝑥𝑋 𝐵𝑆))
32pm5.32i 454 . 2 ((𝐵𝐴𝐵 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
41, 3bitri 184 1 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2160  wral 2468  cin 3143   ciin 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-iin 3904
This theorem is referenced by: (None)
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