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Theorem elriin 4038
Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
elriin (B (Ax 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 3219 . 2 (B (Ax X S) ↔ (B A B x X S))
2 eliin 3974 . . 3 (B A → (B x X Sx X B S))
32pm5.32i 618 . 2 ((B A B x X S) ↔ (B A x X B S))
41, 3bitri 240 1 (B (Ax X S) ↔ (B A x X B S))
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358   wcel 1710  wral 2614  cin 3208  ciin 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-iin 3972
This theorem is referenced by: (None)
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