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Theorem rspcimedv 3209
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1  |-  ( ph  ->  A  e.  B )
rspcimedv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
Assertion
Ref Expression
rspcimedv  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Distinct variable groups:    x, A    x, B    ph, x    ch, x
Allowed substitution hint:    ps( x)

Proof of Theorem rspcimedv
StepHypRef Expression
1 rspcimdv.1 . . . 4  |-  ( ph  ->  A  e.  B )
2 rspcimedv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
32con3d 133 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  ->  -.  ch )
)
41, 3rspcimdv 3208 . . 3  |-  ( ph  ->  ( A. x  e.  B  -.  ps  ->  -. 
ch ) )
54con2d 115 . 2  |-  ( ph  ->  ( ch  ->  -.  A. x  e.  B  -.  ps ) )
6 dfrex2 2905 . 2  |-  ( E. x  e.  B  ps  <->  -. 
A. x  e.  B  -.  ps )
75, 6syl6ibr 227 1  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-v 3108
This theorem is referenced by:  rspcedv  3211  scshwfzeqfzo  12785  symgfixfo  16663  slesolex  19351  clwlkfoclwwlk  25047  el2wlkonot  25071  el2spthonot  25072  el2wlkonotot0  25074  usg2spot2nb  25267  usgra2pthlem1  32725
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