Theorem elintab 2548

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 2543	. 2 |- (A e. |^|{x | ph} <-> A.y(y e. {x | ph} -> A e. y))
3		hbab1 1469	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
4		ax-17 973	. . . 4 |- (A e. y -> A.x A e. y)
5	3, 4	hbim 1009	. . 3 |- ((y e. {x | ph} -> A e. y) -> A.x(y e. {x | ph} -> A e. y))
6		ax-17 973	. . 3 |- ((ph -> A e. x) -> A.y(ph -> A e. x))
7		eleq1 1537	. . . . 5 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
8		abid 1468	. . . . 5 |- (x e. {x | ph} <-> ph)
9	7, 8	syl6bb 538	. . . 4 |- (y = x -> (y e. {x | ph} <-> ph))
10		eleq2 1538	. . . 4 |- (y = x -> (A e. y <-> A e. x))
11	9, 10	imbi12d 628	. . 3 |- (y = x -> ((y e. {x | ph} -> A e. y) <-> (ph -> A e. x)))
12	5, 6, 11	cbval 1167	. 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 956   = wceq 958   e. wcel 960  {cab 1466  Vcvv 1814  |^|cint 2537

This theorem is referenced by:  elintrab 2549  intmin4 2563  intab 2564  dfom3 4639  1nn 5936  peano2nn 5937  dfuz 6204

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-int 2538

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