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Theorem abexssex 6128
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 2461 . . . 4  |-  ( E. x  e.  ~P  A ph 
<->  E. x ( x  e.  ~P A  /\  ph ) )
2 velpw 3584 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
32anbi1i 458 . . . . 5  |-  ( ( x  e.  ~P A  /\  ph )  <->  ( x  C_  A  /\  ph )
)
43exbii 1605 . . . 4  |-  ( E. x ( x  e. 
~P A  /\  ph ) 
<->  E. x ( x 
C_  A  /\  ph ) )
51, 4bitri 184 . . 3  |-  ( E. x  e.  ~P  A ph 
<->  E. x ( x 
C_  A  /\  ph ) )
65abbii 2293 . 2  |-  { y  |  E. x  e. 
~P  A ph }  =  { y  |  E. x ( x  C_  A  /\  ph ) }
7 abrexex2.1 . . . 4  |-  A  e. 
_V
87pwex 4185 . . 3  |-  ~P A  e.  _V
9 abrexex2.2 . . 3  |-  { y  |  ph }  e.  _V
108, 9abrexex2 6127 . 2  |-  { y  |  E. x  e. 
~P  A ph }  e.  _V
116, 10eqeltrri 2251 1  |-  { y  |  E. x ( x  C_  A  /\  ph ) }  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 104   E.wex 1492    e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2739    C_ wss 3131   ~Pcpw 3577
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226
This theorem is referenced by: (None)
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