Theorem elintab 3976

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 3971	. 2 |- ( A e. |^| { x | ph } <-> A. y ( y e. { x | ph } -> A e. y ) )
3		nfsab1 2228	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1581	. . . 4 |- F/ x A e. y
5	3, 4	nfim 1625	. . 3 |- F/ x ( y e. { x | ph } -> A e. y )
6		nfv 1581	. . 3 |- F/ y ( ph -> A e. x )
7		eleq1 2301	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2226	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	bitrdi 196	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10		eleq2 2302	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
11	9, 10	imbi12d 234	. . 3 |- ( y = x -> ( ( y e. { x | ph } -> A e. y ) <-> ( ph -> A e. x ) ) )
12	5, 6, 11	cbval 1807	. 2 |- ( A. y ( y e. { x | ph } -> A e. y ) <-> A. x ( ph -> A e. x ) )
13	2, 12	bitri 184	1 |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   e. wcel 2209   {cab 2224   _Vcvv 2821   |^|cint 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-int 3966

This theorem is referenced by:  elintrab  3977  intmin4  3993  intab  3994  intid  4359