Theorem elintab 2539

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:   x,A

Proof of Theorem elintab

Step	Hyp	Ref	Expression
1		inteqab.1	. . 3 |- A e. V
2	1	elint 2534	. 2 |- (A e. |^|{x | ph} <-> A.y(y e. {x | ph} -> A e. y))
3		hbab1 1464	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
4		ax-17 969	. . . 4 |- (A e. y -> A.x A e. y)
5	3, 4	hbim 1005	. . 3 |- ((y e. {x | ph} -> A e. y) -> A.x(y e. {x | ph} -> A e. y))
6		ax-17 969	. . 3 |- ((ph -> A e. x) -> A.y(ph -> A e. x))
7		eleq1 1531	. . . . 5 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
8		abid 1463	. . . . 5 |- (x e. {x | ph} <-> ph)
9	7, 8	syl6bb 535	. . . 4 |- (y = x -> (y e. {x | ph} <-> ph))
10		eleq2 1532	. . . 4 |- (y = x -> (A e. y <-> A e. x))
11	9, 10	imbi12d 625	. . 3 |- (y = x -> ((y e. {x | ph} -> A e. y) <-> (ph -> A e. x)))
12	5, 6, 11	cbval 1163	. 2 |- (A.y(y e. {x | ph} -> A e. y) <-> A.x(ph -> A e. x))
13	2, 12	bitr 173	1 |- (A e. |^|{x | ph} <-> A.x(ph -> A e. x))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  {cab 1461  Vcvv 1807  |^|cint 2528

This theorem is referenced by:  elintrab 2540  intmin4 2554  intab 2555  dfom3 4610  1nn 5890  peano2nn 5891  dfuz 6158

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-int 2529

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