Theorem elintrab 1977

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(s):   x,A

Proof of Theorem elintrab

Step	Hyp	Ref	Expression
1		inteqab.1	. . . 4 |- A e. V
2	1	elintab 1976	. . 3 |- (A e. |^|{x | (x e. B /\ ph)} <-> A.x((x e. B /\ ph) -> A e. x))
3		impexp 276	. . . 4 |- (((x e. B /\ ph) -> A e. x) <-> (x e. B -> (ph -> A e. x)))
4	3	bial 695	. . 3 |- (A.x((x e. B /\ ph) -> A e. x) <-> A.x(x e. B -> (ph -> A e. x)))
5	2, 4	bitr 151	. 2 |- (A e. |^|{x | (x e. B /\ ph)} <-> A.x(x e. B -> (ph -> A e. x)))
6		df-rab 1208	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
7	6	inteqi 1969	. . 3 |- |^|{x e. B | ph} = |^|{x | (x e. B /\ ph)}
8	7	eleq2i 1153	. 2 |- (A e. |^|{x e. B | ph} <-> A e. |^|{x | (x e. B /\ ph)})
9		df-ral 1205	. 2 |- (A.x e. B (ph -> A e. x) <-> A.x(x e. B -> (ph -> A e. x)))
10	5, 8, 9	3bitr4 158	1 |- (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  {cab 1090   e. wcel 1092  A.wral 1201  {crab 1204  Vcvv 1348  |^|cint 1965

This theorem is referenced by:  intmin 1982  rankun 3535  elspan 5449

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-ral 1205  df-rab 1208  df-v 1349  df-int 1966

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