Theorem elintab 3938

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
inteqab.1	|- A e. _V

Assertion

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

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem elintab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inteqab.1	. . 3 |- A e. _V
2	1	elint 3933	. 2 |- (A e. |^|{x | ph} <-> A.y(y e. {x | ph} -> A e. y))
3		nfsab1 2343	. . . 4 |- F/x y e. {x | ph}
4		nfv 1619	. . . 4 |- F/x A e. y
5	3, 4	nfim 1813	. . 3 |- F/x(y e. {x | ph} -> A e. y)
6		nfv 1619	. . 3 |- F/y(ph -> A e. x)
7		eleq1 2413	. . . . 5 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
8		abid 2341	. . . . 5 |- (x e. {x | ph} <-> ph)
9	7, 8	syl6bb 252	. . . 4 |- (y = x -> (y e. {x | ph} <-> ph))
10		eleq2 2414	. . . 4 |- (y = x -> (A e. y <-> A e. x))
11	9, 10	imbi12d 311	. . 3 |- (y = x -> ((y e. {x | ph} -> A e. y) <-> (ph -> A e. x)))
12	5, 6, 11	cbval 1984	. 2 |- (A.y(y e. {x | ph} -> A e. y) <-> A.x(ph -> A e. x))
13	2, 12	bitri 240	1 |- (A e. |^|{x | ph} <-> A.x(ph -> A e. x))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  _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-v 2862  df-int 3928

This theorem is referenced by:  elintrab  3939  intmin4  3956  intab  3957  peano1  4403  peano2  4404  ncvspfin  4539  spfinsfincl  4540  clos1conn  5880  nchoicelem10  6299