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Theorem cgsexg 2930
Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
Hypotheses
Ref Expression
cgsexg.1  |-  ( x  =  A  ->  ch )
cgsexg.2  |-  ( ch 
->  ( ph  <->  ps )
)
Assertion
Ref Expression
cgsexg  |-  ( A  e.  V  ->  ( E. x ( ch  /\  ph )  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    ch( x)    V( x)

Proof of Theorem cgsexg
StepHypRef Expression
1 cgsexg.2 . . . 4  |-  ( ch 
->  ( ph  <->  ps )
)
21biimpa 471 . . 3  |-  ( ( ch  /\  ph )  ->  ps )
32exlimiv 1641 . 2  |-  ( E. x ( ch  /\  ph )  ->  ps )
4 elisset 2909 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
5 cgsexg.1 . . . . 5  |-  ( x  =  A  ->  ch )
65eximi 1582 . . . 4  |-  ( E. x  x  =  A  ->  E. x ch )
74, 6syl 16 . . 3  |-  ( A  e.  V  ->  E. x ch )
81biimprcd 217 . . . . 5  |-  ( ps 
->  ( ch  ->  ph )
)
98ancld 537 . . . 4  |-  ( ps 
->  ( ch  ->  ( ch  /\  ph ) ) )
109eximdv 1629 . . 3  |-  ( ps 
->  ( E. x ch 
->  E. x ( ch 
/\  ph ) ) )
117, 10syl5com 28 . 2  |-  ( A  e.  V  ->  ( ps  ->  E. x ( ch 
/\  ph ) ) )
123, 11impbid2 196 1  |-  ( A  e.  V  ->  ( E. x ( ch  /\  ph )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-v 2901
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