Theorem elintab 1976

Description: Membership in the intersection of a class abstraction.

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(s):   x,A

Proof of Theorem elintab

Step	Hyp	Ref	Expression
1		inteqab.1	. . 3 |- A e. V
2	1	elint 1971	. 2 |- (A e. |^|{x | ph} <-> A.y(y e. {x | ph} -> A e. y))
3		hbab1 1095	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
4		ax-17 925	. . . 4 |- (A e. y -> A.x A e. y)
5	3, 4	hbim 702	. . 3 |- ((y e. {x | ph} -> A e. y) -> A.x(y e. {x | ph} -> A e. y))
6		ax-17 925	. . 3 |- ((ph -> A e. x) -> A.y(ph -> A e. x))
7		eleq1 1149	. . . . 5 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
8		abid 1094	. . . . 5 |- (x e. {x | ph} <-> ph)
9	7, 8	syl6bb 414	. . . 4 |- (y = x -> (y e. {x | ph} <-> ph))
10		eleq2 1150	. . . 4 |- (y = x -> (A e. y <-> A e. x))
11	9, 10	imbi12d 474	. . 3 |- (y = x -> ((y e. {x | ph} -> A e. y) <-> (ph -> A e. x)))
12	5, 6, 11	cbval 848	. 2 |- (A.y(y e. {x | ph} -> A e. y) <-> A.x(ph -> A e. x))
13	2, 12	bitr 151	1 |- (A e. |^|{x | ph} <-> A.x(ph -> A e. x))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   = weq 797  {cab 1090   e. wcel 1092  Vcvv 1348  |^|cint 1965

This theorem is referenced by:  elintrab 1977  dfom3 3477  1nn 4432  peano2nn 4433

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-int 1966

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