Theorem elintab 4299

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 4294	. 2 |- ( A e. |^| { x | ph } <-> A. y ( y e. { x | ph } -> A e. y ) )
3		nfsab1 2446	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1708	. . . 4 |- F/ x A e. y
5	3, 4	nfim 1921	. . 3 |- F/ x ( y e. { x | ph } -> A e. y )
6		nfv 1708	. . 3 |- F/ y ( ph -> A e. x )
7		eleq1 2529	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2444	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 261	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10		eleq2 2530	. . . 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 2022	. 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 1393   e. wcel 1819   {cab 2442   _Vcvv 3109   |^|cint 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-int 4289

This theorem is referenced by:  elintrab  4300  intmin4  4318  intab  4319  intid  4714  dfom3  8081  dfom5  8084  tc2  8190  dfnn2  10569  efgi  16864  efgi2  16870  mclsax  29126  heibor1lem  30510  brintclab  37923  lem1frege76a  38157