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Theorem elrint 3967
 Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
elrint (X (AB) ↔ (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 3219 . 2 (X (AB) ↔ (X A X B))
2 elintg 3934 . . 3 (X A → (X By B X y))
32pm5.32i 618 . 2 ((X A X B) ↔ (X A y B X y))
41, 3bitri 240 1 (X (AB) ↔ (X A y B X y))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2614   ∩ cin 3208  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-int 3927 This theorem is referenced by:  elrint2  3968
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