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Theorem rspcimdv 2885
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 2548 . 2  |-  ( A. x  e.  B  ps  <->  A. x ( x  e.  B  ->  ps )
)
2 rspcimdv.1 . . 3  |-  ( ph  ->  A  e.  B )
3 simpr 447 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
43eleq1d 2349 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
x  e.  B  <->  A  e.  B ) )
54biimprd 214 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( A  e.  B  ->  x  e.  B ) )
6 rspcimdv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
75, 6imim12d 68 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  B  ->  ps )  ->  ( A  e.  B  ->  ch ) ) )
82, 7spcimdv 2865 . . 3  |-  ( ph  ->  ( A. x ( x  e.  B  ->  ps )  ->  ( A  e.  B  ->  ch ) ) )
92, 8mpid 37 . 2  |-  ( ph  ->  ( A. x ( x  e.  B  ->  ps )  ->  ch )
)
101, 9syl5bi 208 1  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543
This theorem is referenced by:  rspcimedv  2886  rspcdv  2887  mreexd  13544  mreexexlemd  13546  catcocl  13587  catass  13588  moni  13639  subccocl  13719  funcco  13745  fullfo  13786  fthf1  13791  nati  13829  acsfiindd  14280  lmxrge0  23375  funressnfv  27991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790
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