Theorem elrint2 3885

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

Assertion

Ref	Expression
elrint2	|- ( X e. A -> ( X e. ( A i^i |^| B ) <-> A. y e. B X e. y ) )

Distinct variable groups:   y, B   y, X

Allowed substitution hint:   A( y)

Proof of Theorem elrint2

Step	Hyp	Ref	Expression
1		elrint 3884	. 2 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )
2	1	baib 919	1 |- ( X e. A -> ( X e. ( A i^i |^| B ) <-> A. y e. B X e. y ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2148   A.wral 2455   i^i cin 3128   |^|cint 3844

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-int 3845

This theorem is referenced by: (None)