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Theorem abexssex 5797
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 2562 . . . 4  |-  ( E. x  e.  ~P  A ph 
<->  E. x ( x  e.  ~P A  /\  ph ) )
2 abrexex2.1 . . . . . . 7  |-  A  e. 
_V
32elpw2 4191 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
43anbi1i 676 . . . . 5  |-  ( ( x  e.  ~P A  /\  ph )  <->  ( x  C_  A  /\  ph )
)
54exbii 1572 . . . 4  |-  ( E. x ( x  e. 
~P A  /\  ph ) 
<->  E. x ( x 
C_  A  /\  ph ) )
61, 5bitri 240 . . 3  |-  ( E. x  e.  ~P  A ph 
<->  E. x ( x 
C_  A  /\  ph ) )
76abbii 2408 . 2  |-  { y  |  E. x  e. 
~P  A ph }  =  { y  |  E. x ( x  C_  A  /\  ph ) }
82pwex 4209 . . 3  |-  ~P A  e.  _V
9 abrexex2.2 . . 3  |-  { y  |  ph }  e.  _V
108, 9abrexex2 5796 . 2  |-  { y  |  E. x  e. 
~P  A ph }  e.  _V
117, 10eqeltrri 2367 1  |-  { y  |  E. x ( x  C_  A  /\  ph ) }  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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