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

Theorem elunirab 3621
Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
Assertion
Ref Expression
elunirab  |-  ( A  e.  U. { x  e.  B  |  ph }  <->  E. x  e.  B  ( A  e.  x  /\  ph ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elunirab
StepHypRef Expression
1 eluniab 3620 . 2  |-  ( A  e.  U. { x  |  ( x  e.  B  /\  ph ) } 
<->  E. x ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
2 df-rab 2332 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
32unieqi 3618 . . 3  |-  U. {
x  e.  B  |  ph }  =  U. {
x  |  ( x  e.  B  /\  ph ) }
43eleq2i 2120 . 2  |-  ( A  e.  U. { x  e.  B  |  ph }  <->  A  e.  U. { x  |  ( x  e.  B  /\  ph ) } )
5 df-rex 2329 . . 3  |-  ( E. x  e.  B  ( A  e.  x  /\  ph )  <->  E. x ( x  e.  B  /\  ( A  e.  x  /\  ph ) ) )
6 an12 503 . . . 4  |-  ( ( x  e.  B  /\  ( A  e.  x  /\  ph ) )  <->  ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
76exbii 1512 . . 3  |-  ( E. x ( x  e.  B  /\  ( A  e.  x  /\  ph ) )  <->  E. x
( A  e.  x  /\  ( x  e.  B  /\  ph ) ) )
85, 7bitri 177 . 2  |-  ( E. x  e.  B  ( A  e.  x  /\  ph )  <->  E. x ( A  e.  x  /\  (
x  e.  B  /\  ph ) ) )
91, 4, 83bitr4i 205 1  |-  ( A  e.  U. { x  e.  B  |  ph }  <->  E. x  e.  B  ( A  e.  x  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324   {crab 2327   U.cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-rab 2332  df-v 2576  df-uni 3609
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator