Theorem elintrab 2542

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 2541	. . 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 998	. . 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 1651	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
7	6	inteqi 2534	. . 3 |- |^|{x e. B | ph} = |^|{x | (x e. B /\ ph)}
8	7	eleq2i 1537	. 2 |- (A e. |^|{x e. B | ph} <-> A e. |^|{x | (x e. B /\ ph)})
9		df-ral 1648	. 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 953   e. wcel 957  {cab 1463  A.wral 1644  {crab 1647  Vcvv 1809  |^|cint 2530

This theorem is referenced by:  elintrabg 2543  intmin 2550  rankun 4678  clsval2 7664  elspan 9454

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1648  df-rab 1651  df-v 1810  df-int 2531

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