Theorem elintrab 4488

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)

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

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem elintrab

Step	Hyp	Ref	Expression
1		inteqab.1	. . . 4 |- A e. _V
2	1	elintab 4487	. . 3 |- ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A e. x ) )
3		impexp 462	. . . 4 |- ( ( ( x e. B /\ ph ) -> A e. x ) <-> ( x e. B -> ( ph -> A e. x ) ) )
4	3	albii 1747	. . 3 |- ( A. x ( ( x e. B /\ ph ) -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
5	2, 4	bitri 264	. 2 |- ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
6		df-rab 2921	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
7	6	inteqi 4479	. . 3 |- |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
8	7	eleq2i 2693	. 2 |- ( A e. |^| { x e. B | ph } <-> A e. |^| { x | ( x e. B /\ ph ) } )
9		df-ral 2917	. 2 |- ( A. x e. B ( ph -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
10	5, 8, 9	3bitr4i 292	1 |- ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-int 4476

This theorem is referenced by:  elintrabg  4489  intmin  4497  rankunb  8713  isf34lem4  9199  ist1-3  21153  filufint  21724  elspani  28402  ldsysgenld  30223  ldgenpisyslem1  30226  kur14lem9  31196  pclclN  35177  elpclN  35178  lcosslsp  42227