Theorem elintab 4245

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 4240	. 2 |- ( A e. |^| { x | ph } <-> A. y ( y e. { x | ph } -> A e. y ) )
3		nfsab1 2441	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1761	. . . 4 |- F/ x A e. y
5	3, 4	nfim 2003	. . 3 |- F/ x ( y e. { x | ph } -> A e. y )
6		nfv 1761	. . 3 |- F/ y ( ph -> A e. x )
7		eleq1 2517	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2439	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 265	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10		eleq2 2518	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
11	9, 10	imbi12d 322	. . 3 |- ( y = x -> ( ( y e. { x | ph } -> A e. y ) <-> ( ph -> A e. x ) ) )
12	5, 6, 11	cbval 2114	. 2 |- ( A. y ( y e. { x | ph } -> A e. y ) <-> A. x ( ph -> A e. x ) )
13	2, 12	bitri 253	1 |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 188   A.wal 1442   e. wcel 1887   {cab 2437   _Vcvv 3045   |^|cint 4234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-int 4235

This theorem is referenced by:  elintrab  4246  intmin4  4264  intab  4265  intid  4658  dfom3  8152  dfom5  8155  tc2  8226  dfnn2  10622  brintclab  13065  efgi  17369  efgi2  17375  mclsax  30207  heibor1lem  32141  elmapintab  36202  intabssd  36216  cotrintab  36221  dffrege76  36535