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Theorem spcimedv 3051
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1  |-  ( ph  ->  A  e.  B )
spcimedv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
Assertion
Ref Expression
spcimedv  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    B( x)

Proof of Theorem spcimedv
StepHypRef Expression
1 spcimdv.1 . . . 4  |-  ( ph  ->  A  e.  B )
2 spcimedv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
32con3d 133 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  ->  -.  ch )
)
41, 3spcimdv 3049 . . 3  |-  ( ph  ->  ( A. x  -.  ps  ->  -.  ch )
)
54con2d 115 . 2  |-  ( ph  ->  ( ch  ->  -.  A. x  -.  ps )
)
6 df-ex 1587 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
75, 6syl6ibr 227 1  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969
This theorem is referenced by:  hashf1rn  12115  cshwsexa  12450  uvcendim  18251  wwlktovfo  30206  wlkiswwlk2  30284  wlknwwlknsur  30297  wlkiswwlksur  30304  wwlkextsur  30316  clwlkisclwwlklem2  30401  clwlksizeeq  30478
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