Theorem elrint 3914

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 3346	. 2 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ X e. |^| B ) )
2		elintg 3882	. . 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 2167   A.wral 2475   i^i cin 3156   |^|cint 3874

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-int 3875

This theorem is referenced by:  elrint2  3915