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Theorem elriin 3718
 Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
elriin (B (A x 𝑋 𝑆) ↔ (B A x 𝑋 B 𝑆))
Distinct variable groups:   x,A   x,𝑋   x,B
Allowed substitution hint:   𝑆(x)

Proof of Theorem elriin
StepHypRef Expression
1 elin 3120 . 2 (B (A x 𝑋 𝑆) ↔ (B A B x 𝑋 𝑆))
2 eliin 3653 . . 3 (B A → (B x 𝑋 𝑆x 𝑋 B 𝑆))
32pm5.32i 427 . 2 ((B A B x 𝑋 𝑆) ↔ (B A x 𝑋 B 𝑆))
41, 3bitri 173 1 (B (A x 𝑋 𝑆) ↔ (B A x 𝑋 B 𝑆))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∀wral 2300   ∩ cin 2910  ∩ ciin 3649 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-iin 3651 This theorem is referenced by: (None)
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