Theorem elrint 3811

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 3259	. 2 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ X e. |^| B ) )
2		elintg 3779	. . 3 |- ( X e. A -> ( X e. |^| B <-> A. y e. B X e. y ) )
3	2	pm5.32i 449	. 2 |- ( ( X e. A /\ X e. |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )
4	1, 3	bitri 183	1 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1480   A.wral 2416   i^i cin 3070   |^|cint 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-int 3772

This theorem is referenced by:  elrint2  3812