Theorem elintrab 3228

Description: Membership in the intersection of a class abstraction.

Hypothesis

Ref	Expression
inteqab.1	|- A e. _V

Assertion

Ref	Expression
elintrab	|- (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x))

Distinct variable group:   x,A

Proof of Theorem elintrab

Step	Hyp	Ref	Expression
1		inteqab.1	. . . 4 |- A e. _V
2	1	elintab 3227	. . 3 |- (A e. |^|{x | (x e. B /\ ph)} <-> A.x((x e. B /\ ph) -> A e. x))
3		impexp 374	. . . 4 |- (((x e. B /\ ph) -> A e. x) <-> (x e. B -> (ph -> A e. x)))
4	3	albii 1346	. . 3 |- (A.x((x e. B /\ ph) -> A e. x) <-> A.x(x e. B -> (ph -> A e. x)))
5	2, 4	bitri 190	. 2 |- (A e. |^|{x | (x e. B /\ ph)} <-> A.x(x e. B -> (ph -> A e. x)))
6		df-rab 2112	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
7	6	inteqi 3218	. . 3 |- |^|{x e. B | ph} = |^|{x | (x e. B /\ ph)}
8	7	eleq2i 1961	. 2 |- (A e. |^|{x e. B | ph} <-> A e. |^|{x | (x e. B /\ ph)})
9		df-ral 2109	. 2 |- (A.x e. B (ph -> A e. x) <-> A.x(x e. B -> (ph -> A e. x)))
10	5, 8, 9	3bitr4i 200	1 |- (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  {cab 1871  A.wral 2105  {crab 2108  _Vcvv 2292  |^|cint 3214

This theorem is referenced by:  elintrabg 3229  intmin 3237  intminOLD 3238  rankun 5802  clsval2 8961  elspani 11099  ist1-3 15543  filufint 15574

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rab 2112  df-v 2294  df-int 3215

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