Theorem elintab 3699

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 3694	. 2 |- ( A e. |^| { x | ph } <-> A. y ( y e. { x | ph } -> A e. y ) )
3		nfsab1 2078	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1466	. . . 4 |- F/ x A e. y
5	3, 4	nfim 1509	. . 3 |- F/ x ( y e. { x | ph } -> A e. y )
6		nfv 1466	. . 3 |- F/ y ( ph -> A e. x )
7		eleq1 2150	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2076	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 194	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10		eleq2 2151	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
11	9, 10	imbi12d 232	. . 3 |- ( y = x -> ( ( y e. { x | ph } -> A e. y ) <-> ( ph -> A e. x ) ) )
12	5, 6, 11	cbval 1684	. 2 |- ( A. y ( y e. { x | ph } -> A e. y ) <-> A. x ( ph -> A e. x ) )
13	2, 12	bitri 182	1 |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1287   e. wcel 1438   {cab 2074   _Vcvv 2619   |^|cint 3688

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-int 3689

This theorem is referenced by:  elintrab  3700  intmin4  3716  intab  3717  intid  4051