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Theorem elrint 3646
Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
elrint (𝑋 (A B) ↔ (𝑋 A y B 𝑋 y))
Distinct variable groups:   y,B   y,𝑋
Allowed substitution hint:   A(y)

Proof of Theorem elrint
StepHypRef Expression
1 elin 3120 . 2 (𝑋 (A B) ↔ (𝑋 A 𝑋 B))
2 elintg 3614 . . 3 (𝑋 A → (𝑋 By B 𝑋 y))
32pm5.32i 427 . 2 ((𝑋 A 𝑋 B) ↔ (𝑋 A y B 𝑋 y))
41, 3bitri 173 1 (𝑋 (A B) ↔ (𝑋 A y B 𝑋 y))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wcel 1390  wral 2300  cin 2910   cint 3606
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-int 3607
This theorem is referenced by:  elrint2  3647
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