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Theorem cgsexg 2819
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 470 . . 3  |-  ( ( ch  /\  ph )  ->  ps )
32exlimiv 1666 . 2  |-  ( E. x ( ch  /\  ph )  ->  ps )
4 elisset 2798 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
5 cgsexg.1 . . . . 5  |-  ( x  =  A  ->  ch )
65eximi 1563 . . . 4  |-  ( E. x  x  =  A  ->  E. x ch )
74, 6syl 15 . . 3  |-  ( A  e.  V  ->  E. x ch )
81biimprcd 216 . . . . 5  |-  ( ps 
->  ( ch  ->  ph )
)
98ancld 536 . . . 4  |-  ( ps 
->  ( ch  ->  ( ch  /\  ph ) ) )
109eximdv 1608 . . 3  |-  ( ps 
->  ( E. x ch 
->  E. x ( ch 
/\  ph ) ) )
117, 10syl5com 26 . 2  |-  ( A  e.  V  ->  ( ps  ->  E. x ( ch 
/\  ph ) ) )
123, 11impbid2 195 1  |-  ( A  e.  V  ->  ( E. x ( ch  /\  ph )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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