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Theorem elriin 3724
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 3123 . 2 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴𝐵 𝑥𝑋 𝑆))
2 eliin 3659 . . 3 (𝐵𝐴 → (𝐵 𝑥𝑋 𝑆 ↔ ∀𝑥𝑋 𝐵𝑆))
32pm5.32i 427 . 2 ((𝐵𝐴𝐵 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
41, 3bitri 173 1 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wcel 1393  wral 2303  cin 2913   ciin 3655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-in 2921  df-iin 3657
This theorem is referenced by: (None)
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