Theorem elintrab 2545

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 2544	. . 3 |- (A e. |^|{x | (x e. B /\ ph)} <-> A.x((x e. B /\ ph) -> A e. x))
3		impexp 347	. . . 4 |- (((x e. B /\ ph) -> A e. x) <-> (x e. B -> (ph -> A e. x)))
4	3	albii 999	. . 3 |- (A.x((x e. B /\ ph) -> A e. x) <-> A.x(x e. B -> (ph -> A e. x)))
5	2, 4	bitr 173	. 2 |- (A e. |^|{x | (x e. B /\ ph)} <-> A.x(x e. B -> (ph -> A e. x)))
6		df-rab 1652	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
7	6	inteqi 2537	. . 3 |- |^|{x e. B | ph} = |^|{x | (x e. B /\ ph)}
8	7	eleq2i 1538	. 2 |- (A e. |^|{x e. B | ph} <-> A e. |^|{x | (x e. B /\ ph)})
9		df-ral 1649	. 2 |- (A.x e. B (ph -> A e. x) <-> A.x(x e. B -> (ph -> A e. x)))
10	5, 8, 9	3bitr4 183	1 |- (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   e. wcel 958  {cab 1463  A.wral 1645  {crab 1648  Vcvv 1811  |^|cint 2533

This theorem is referenced by:  elintrabg 2546  intmin 2553  rankun 4691  clsval2 7685  elspan 9466

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rab 1652  df-v 1812  df-int 2534

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