Theorem elrint2 3969

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 3968	. 2 |- (X e. (A i^i |^|B) <-> (X e. A /\ A.y e. B X e. y))
2	1	baib 871	1 |- (X e. A -> (X e. (A i^i |^|B) <-> A.y e. B X e. y))

Syntax hints:   -> wi 4   <-> wb 176   e. wcel 1710  A.wral 2615   i^i cin 3209  |^|cint 3927

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 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-int 3928

This theorem is referenced by: (None)