Theorem elrint 3925

Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
elrint	|- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )

Distinct variable groups:   y, B   y, X

Allowed substitution hint:   A( y)

Proof of Theorem elrint

Step	Hyp	Ref	Expression
1		elin 3356	. 2 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ X e. |^| B ) )
2		elintg 3893	. . 3 |- ( X e. A -> ( X e. |^| B <-> A. y e. B X e. y ) )
3	2	pm5.32i 454	. 2 |- ( ( X e. A /\ X e. |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )
4	1, 3	bitri 184	1 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2176   A.wral 2484   i^i cin 3165   |^|cint 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-int 3886

This theorem is referenced by:  elrint2  3926