Theorem elintab 4240

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 4235	. 2 |- ( A e. |^| { x | ph } <-> A. y ( y e. { x | ph } -> A e. y ) )
3		nfsab1 2440	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1674	. . . 4 |- F/ x A e. y
5	3, 4	nfim 1855	. . 3 |- F/ x ( y e. { x | ph } -> A e. y )
6		nfv 1674	. . 3 |- F/ y ( ph -> A e. x )
7		eleq1 2523	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2438	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	syl6bb 261	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10		eleq2 2524	. . . 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 1978	. 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 1368   e. wcel 1758   {cab 2436   _Vcvv 3071   |^|cint 4229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073  df-int 4230

This theorem is referenced by:  elintrab  4241  intmin4  4258  intab  4259  intid  4651  dfom3  7957  dfom5  7960  tc2  8066  dfnn2  10439  efgi  16329  efgi2  16335  heibor1lem  28849