Theorem elint2 3956

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Hypothesis

Ref	Expression
elint2.1	|- A e. _V

Assertion

Ref	Expression
elint2	|- ( A e. |^| B <-> A. x e. B A e. x )

Distinct variable groups:   x, A   x, B

Proof of Theorem elint2

Step	Hyp	Ref	Expression
1		elint2.1	. . 3 |- A e. _V
2	1	elint 3955	. 2 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )
3		df-ral 2525	. 2 |- ( A. x e. B A e. x <-> A. x ( x e. B -> A e. x ) )
4	2, 3	bitr4i 187	1 |- ( A e. |^| B <-> A. x e. B A e. x )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   e. wcel 2203   A.wral 2520   _Vcvv 2813   |^|cint 3949

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-int 3950

This theorem is referenced by:  elintg  3957  ssint  3965  intssunim  3971  iinuniss  4074  trint  4223  suplocexprlemmu  8033  suplocexprlemdisj  8035  suplocexprlemloc  8036  suplocexprlemub  8038