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Theorem rspcimdv 3079
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1  |-  ( ph  ->  A  e.  B )
rspcimdv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
rspcimdv  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Distinct variable groups:    x, A    x, B    ph, x    ch, x
Allowed substitution hint:    ps( x)

Proof of Theorem rspcimdv
StepHypRef Expression
1 df-ral 2725 . 2  |-  ( A. x  e.  B  ps  <->  A. x ( x  e.  B  ->  ps )
)
2 rspcimdv.1 . . 3  |-  ( ph  ->  A  e.  B )
3 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
43eleq1d 2509 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
x  e.  B  <->  A  e.  B ) )
54biimprd 223 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( A  e.  B  ->  x  e.  B ) )
6 rspcimdv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
75, 6imim12d 74 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  B  ->  ps )  ->  ( A  e.  B  ->  ch ) ) )
82, 7spcimdv 3059 . . 3  |-  ( ph  ->  ( A. x ( x  e.  B  ->  ps )  ->  ( A  e.  B  ->  ch ) ) )
92, 8mpid 41 . 2  |-  ( ph  ->  ( A. x ( x  e.  B  ->  ps )  ->  ch )
)
101, 9syl5bi 217 1  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756   A.wral 2720
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 2423
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-v 2979
This theorem is referenced by:  rspcimedv  3080  rspcdv  3081  wrd2ind  12377  mreexd  14585  mreexexlemd  14587  catcocl  14628  catass  14629  moni  14680  subccocl  14760  funcco  14786  fullfo  14827  fthf1  14832  nati  14870  acsfiindd  15352  sizeusglecusglem1  23397  lmxrge0  26387  funressnfv  30039  friendshipgt3  30719  snlindsntorlem  31009
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