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Theorem cgsexg 2721
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 294 . . 3  |-  ( ( ch  /\  ph )  ->  ps )
32exlimiv 1577 . 2  |-  ( E. x ( ch  /\  ph )  ->  ps )
4 elisset 2700 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
5 cgsexg.1 . . . . 5  |-  ( x  =  A  ->  ch )
65eximi 1579 . . . 4  |-  ( E. x  x  =  A  ->  E. x ch )
74, 6syl 14 . . 3  |-  ( A  e.  V  ->  E. x ch )
81biimprcd 159 . . . . 5  |-  ( ps 
->  ( ch  ->  ph )
)
98ancld 323 . . . 4  |-  ( ps 
->  ( ch  ->  ( ch  /\  ph ) ) )
109eximdv 1852 . . 3  |-  ( ps 
->  ( E. x ch 
->  E. x ( ch 
/\  ph ) ) )
117, 10syl5com 29 . 2  |-  ( A  e.  V  ->  ( ps  ->  E. x ( ch 
/\  ph ) ) )
123, 11impbid2 142 1  |-  ( A  e.  V  ->  ( E. x ( ch  /\  ph )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480
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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688
This theorem is referenced by: (None)
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