MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elintab Unicode version

Theorem elintab 4025
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
inteqab.1  |-  A  e. 
_V
Assertion
Ref Expression
elintab  |-  ( A  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  e.  x ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem elintab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 inteqab.1 . . 3  |-  A  e. 
_V
21elint 4020 . 2  |-  ( A  e.  |^| { x  | 
ph }  <->  A. y
( y  e.  {
x  |  ph }  ->  A  e.  y ) )
3 nfsab1 2398 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1626 . . . 4  |-  F/ x  A  e.  y
53, 4nfim 1828 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  ->  A  e.  y )
6 nfv 1626 . . 3  |-  F/ y ( ph  ->  A  e.  x )
7 eleq1 2468 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2396 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
97, 8syl6bb 253 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
10 eleq2 2469 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
119, 10imbi12d 312 . . 3  |-  ( y  =  x  ->  (
( y  e.  {
x  |  ph }  ->  A  e.  y )  <-> 
( ph  ->  A  e.  x ) ) )
125, 6, 11cbval 2040 . 2  |-  ( A. y ( y  e. 
{ x  |  ph }  ->  A  e.  y )  <->  A. x ( ph  ->  A  e.  x ) )
132, 12bitri 241 1  |-  ( A  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    e. wcel 1721   {cab 2394   _Vcvv 2920   |^|cint 4014
This theorem is referenced by:  elintrab  4026  intmin4  4043  intab  4044  intid  4385  dfom3  7562  dfom5  7565  tc2  7641  dfnn2  9973  efgi  15310  efgi2  15316  heibor1lem  26412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-int 4015
  Copyright terms: Public domain W3C validator