Theorem elintrab 3939

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)

Hypothesis

Ref	Expression
inteqab.1	|- A e. _V

Assertion

Ref	Expression
elintrab	|- (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x))

Distinct variable group:   x,A

Allowed substitution hints:   ph(x)   B(x)

Proof of Theorem elintrab

Step	Hyp	Ref	Expression
1		inteqab.1	. . . 4 |- A e. _V
2	1	elintab 3938	. . 3 |- (A e. |^|{x | (x e. B /\ ph)} <-> A.x((x e. B /\ ph) -> A e. x))
3		impexp 433	. . . 4 |- (((x e. B /\ ph) -> A e. x) <-> (x e. B -> (ph -> A e. x)))
4	3	albii 1566	. . 3 |- (A.x((x e. B /\ ph) -> A e. x) <-> A.x(x e. B -> (ph -> A e. x)))
5	2, 4	bitri 240	. 2 |- (A e. |^|{x | (x e. B /\ ph)} <-> A.x(x e. B -> (ph -> A e. x)))
6		df-rab 2624	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
7	6	inteqi 3931	. . 3 |- |^|{x e. B | ph} = |^|{x | (x e. B /\ ph)}
8	7	eleq2i 2417	. 2 |- (A e. |^|{x e. B | ph} <-> A e. |^|{x | (x e. B /\ ph)})
9		df-ral 2620	. 2 |- (A.x e. B (ph -> A e. x) <-> A.x(x e. B -> (ph -> A e. x)))
10	5, 8, 9	3bitr4i 268	1 |- (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  {cab 2339  A.wral 2615  {crab 2619  _Vcvv 2860  |^|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-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-rab 2624  df-v 2862  df-int 3928

This theorem is referenced by:  elintrabg  3940  intmin  3947