ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  clelab Unicode version

Theorem clelab 2303
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
clelab  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem clelab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-clab 2164 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
21anbi2i 457 . . 3  |-  ( ( y  =  A  /\  y  e.  { x  |  ph } )  <->  ( y  =  A  /\  [ y  /  x ] ph ) )
32exbii 1605 . 2  |-  ( E. y ( y  =  A  /\  y  e. 
{ x  |  ph } )  <->  E. y
( y  =  A  /\  [ y  /  x ] ph ) )
4 df-clel 2173 . 2  |-  ( A  e.  { x  | 
ph }  <->  E. y
( y  =  A  /\  y  e.  {
x  |  ph }
) )
5 nfv 1528 . . 3  |-  F/ y ( x  =  A  /\  ph )
6 nfv 1528 . . . 4  |-  F/ x  y  =  A
7 nfs1v 1939 . . . 4  |-  F/ x [ y  /  x ] ph
86, 7nfan 1565 . . 3  |-  F/ x
( y  =  A  /\  [ y  /  x ] ph )
9 eqeq1 2184 . . . 4  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
10 sbequ12 1771 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
119, 10anbi12d 473 . . 3  |-  ( x  =  y  ->  (
( x  =  A  /\  ph )  <->  ( y  =  A  /\  [ y  /  x ] ph ) ) )
125, 8, 11cbvex 1756 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  A  /\  [ y  /  x ] ph ) )
133, 4, 123bitr4i 212 1  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492   [wsb 1762    e. wcel 2148   {cab 2163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173
This theorem is referenced by:  elrabi  2890
  Copyright terms: Public domain W3C validator