Theorem elintrab 3627

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 3626	. . 3 |- ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A e. x ) )
3		impexp 250	. . . 4 |- ( ( ( x e. B /\ ph ) -> A e. x ) <-> ( x e. B -> ( ph -> A e. x ) ) )
4	3	albii 1359	. . 3 |- ( A. x ( ( x e. B /\ ph ) -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
5	2, 4	bitri 173	. 2 |- ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
6		df-rab 2315	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
7	6	inteqi 3619	. . 3 |- |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
8	7	eleq2i 2104	. 2 |- ( A e. |^| { x e. B | ph } <-> A e. |^| { x | ( x e. B /\ ph ) } )
9		df-ral 2311	. 2 |- ( A. x e. B ( ph -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
10	5, 8, 9	3bitr4i 201	1 |- ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   e. wcel 1393   {cab 2026   A.wral 2306   {crab 2310   _Vcvv 2557   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-int 3616

This theorem is referenced by:  elintrabg  3628  intmin  3635  bj-indint  10055