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Theorem abexssex 5780
Description: Existence of a class abstraction with an existentially quantified expression. Both  x and  y can be free in  ph. (Contributed by NM, 29-Jul-2006.)
Hypotheses
Ref Expression
abrexex2.1  |-  A  e. 
_V
abrexex2.2  |-  { y  |  ph }  e.  _V
Assertion
Ref Expression
abexssex  |-  { y  |  E. x ( x  C_  A  /\  ph ) }  e.  _V
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem abexssex
StepHypRef Expression
1 df-rex 2329 . . . 4  |-  ( E. x  e.  ~P  A ph 
<->  E. x ( x  e.  ~P A  /\  ph ) )
2 selpw 3394 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
32anbi1i 439 . . . . 5  |-  ( ( x  e.  ~P A  /\  ph )  <->  ( x  C_  A  /\  ph )
)
43exbii 1512 . . . 4  |-  ( E. x ( x  e. 
~P A  /\  ph ) 
<->  E. x ( x 
C_  A  /\  ph ) )
51, 4bitri 177 . . 3  |-  ( E. x  e.  ~P  A ph 
<->  E. x ( x 
C_  A  /\  ph ) )
65abbii 2169 . 2  |-  { y  |  E. x  e. 
~P  A ph }  =  { y  |  E. x ( x  C_  A  /\  ph ) }
7 abrexex2.1 . . . 4  |-  A  e. 
_V
87pwex 3960 . . 3  |-  ~P A  e.  _V
9 abrexex2.2 . . 3  |-  { y  |  ph }  e.  _V
108, 9abrexex2 5779 . 2  |-  { y  |  E. x  e. 
~P  A ph }  e.  _V
116, 10eqeltrri 2127 1  |-  { y  |  E. x ( x  C_  A  /\  ph ) }  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 101   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387
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-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938
This theorem is referenced by: (None)
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