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Theorem abexssex 5989
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 2397 . . . 4  |-  ( E. x  e.  ~P  A ph 
<->  E. x ( x  e.  ~P A  /\  ph ) )
2 velpw 3485 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
32anbi1i 451 . . . . 5  |-  ( ( x  e.  ~P A  /\  ph )  <->  ( x  C_  A  /\  ph )
)
43exbii 1567 . . . 4  |-  ( E. x ( x  e. 
~P A  /\  ph ) 
<->  E. x ( x 
C_  A  /\  ph ) )
51, 4bitri 183 . . 3  |-  ( E. x  e.  ~P  A ph 
<->  E. x ( x 
C_  A  /\  ph ) )
65abbii 2231 . 2  |-  { y  |  E. x  e. 
~P  A ph }  =  { y  |  E. x ( x  C_  A  /\  ph ) }
7 abrexex2.1 . . . 4  |-  A  e. 
_V
87pwex 4075 . . 3  |-  ~P A  e.  _V
9 abrexex2.2 . . 3  |-  { y  |  ph }  e.  _V
108, 9abrexex2 5988 . 2  |-  { y  |  E. x  e. 
~P  A ph }  e.  _V
116, 10eqeltrri 2189 1  |-  { y  |  E. x ( x  C_  A  /\  ph ) }  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 103   E.wex 1451    e. wcel 1463   {cab 2101   E.wrex 2392   _Vcvv 2658    C_ wss 3039   ~Pcpw 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099
This theorem is referenced by: (None)
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