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Theorem spcimdv 2865
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1  |-  ( ph  ->  A  e.  B )
spcimdv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
spcimdv  |-  ( ph  ->  ( A. x ps 
->  ch ) )
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    B( x)

Proof of Theorem spcimdv
StepHypRef Expression
1 spcimdv.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
21ex 423 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  ->  ch ) ) )
32alrimiv 1617 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps  ->  ch ) ) )
4 spcimdv.1 . 2  |-  ( ph  ->  A  e.  B )
5 nfv 1605 . . 3  |-  F/ x ch
6 nfcv 2419 . . 3  |-  F/_ x A
75, 6spcimgft 2859 . 2  |-  ( A. x ( x  =  A  ->  ( ps  ->  ch ) )  -> 
( A  e.  B  ->  ( A. x ps 
->  ch ) ) )
83, 4, 7sylc 56 1  |-  ( ph  ->  ( A. x ps 
->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684
This theorem is referenced by:  spcdv  2866  spcimedv  2867  rspcimdv  2885  mrieqv2d  13541  mreexexlemd  13546
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-v 2790
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