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Theorem rspcimdv 2989
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 2647 . 2  |-  ( A. x  e.  B  ps  <->  A. x ( x  e.  B  ->  ps )
)
2 rspcimdv.1 . . 3  |-  ( ph  ->  A  e.  B )
3 simpr 448 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
43eleq1d 2446 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
x  e.  B  <->  A  e.  B ) )
54biimprd 215 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( A  e.  B  ->  x  e.  B ) )
6 rspcimdv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
75, 6imim12d 70 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  B  ->  ps )  ->  ( A  e.  B  ->  ch ) ) )
82, 7spcimdv 2969 . . 3  |-  ( ph  ->  ( A. x ( x  e.  B  ->  ps )  ->  ( A  e.  B  ->  ch ) ) )
92, 8mpid 39 . 2  |-  ( ph  ->  ( A. x ( x  e.  B  ->  ps )  ->  ch )
)
101, 9syl5bi 209 1  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   A.wral 2642
This theorem is referenced by:  rspcimedv  2990  rspcdv  2991  mreexd  13787  mreexexlemd  13789  catcocl  13830  catass  13831  moni  13882  subccocl  13962  funcco  13988  fullfo  14029  fthf1  14034  nati  14072  acsfiindd  14523  sizeusglecusglem1  21352  lmxrge0  24134  funressnfv  27654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-v 2894
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