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Theorem rspcimedv 3056
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 128 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  ->  -.  ch )
)
41, 3rspcimdv 3055 . . 3  |-  ( ph  ->  ( A. x  e.  B  -.  ps  ->  -. 
ch ) )
54con2d 110 . 2  |-  ( ph  ->  ( ch  ->  -.  A. x  e.  B  -.  ps ) )
6 dfrex2 2720 . 2  |-  ( E. x  e.  B  ps  <->  -. 
A. x  e.  B  -.  ps )
75, 6syl6ibr 220 1  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708
This theorem is referenced by:  rspcedv  3058  fargshiftfo  21627  usgra2pthlem1  28336  el2wlkonot  28389  el2spthonot  28390  el2wlkonotot0  28392  usg2spot2nb  28516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960
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