MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cgsexg Unicode version

Theorem cgsexg 2832
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 1624 . 2  |-  ( E. x ( ch  /\  ph )  ->  ps )
4 elisset 2811 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
5 cgsexg.1 . . . . 5  |-  ( x  =  A  ->  ch )
65eximi 1566 . . . 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 1612 . . 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 1531    = wceq 1632    e. wcel 1696
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-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
  Copyright terms: Public domain W3C validator