Theorem elintab 3782

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 3777	. 2 |- ( A e. |^| { x | ph } <-> A. y ( y e. { x | ph } -> A e. y ) )
3		nfsab1 2129	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1508	. . . 4 |- F/ x A e. y
5	3, 4	nfim 1551	. . 3 |- F/ x ( y e. { x | ph } -> A e. y )
6		nfv 1508	. . 3 |- F/ y ( ph -> A e. x )
7		eleq1 2202	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2127	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 195	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10		eleq2 2203	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
11	9, 10	imbi12d 233	. . 3 |- ( y = x -> ( ( y e. { x | ph } -> A e. y ) <-> ( ph -> A e. x ) ) )
12	5, 6, 11	cbval 1727	. 2 |- ( A. y ( y e. { x | ph } -> A e. y ) <-> A. x ( ph -> A e. x ) )
13	2, 12	bitri 183	1 |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   e. wcel 1480   {cab 2125   _Vcvv 2686   |^|cint 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-int 3772

This theorem is referenced by:  elintrab  3783  intmin4  3799  intab  3800  intid  4146