Theorem elintab 4127

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 4122	. 2 |- ( A e. |^| { x | ph } <-> A. y ( y e. { x | ph } -> A e. y ) )
3		nfsab1 2423	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1672	. . . 4 |- F/ x A e. y
5	3, 4	nfim 1851	. . 3 |- F/ x ( y e. { x | ph } -> A e. y )
6		nfv 1672	. . 3 |- F/ y ( ph -> A e. x )
7		eleq1 2493	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2421	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 261	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10		eleq2 2494	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
11	9, 10	imbi12d 320	. . 3 |- ( y = x -> ( ( y e. { x | ph } -> A e. y ) <-> ( ph -> A e. x ) ) )
12	5, 6, 11	cbval 1968	. 2 |- ( A. y ( y e. { x | ph } -> A e. y ) <-> A. x ( ph -> A e. x ) )
13	2, 12	bitri 249	1 |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1360   e. wcel 1755   {cab 2419   _Vcvv 2962   |^|cint 4116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-v 2964  df-int 4117

This theorem is referenced by:  elintrab  4128  intmin4  4145  intab  4146  intid  4538  dfom3  7841  dfom5  7844  tc2  7950  dfnn2  10322  efgi  16195  efgi2  16201  heibor1lem  28549