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Theorem spcimedv 2838
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1  |-  ( ph  ->  A  e.  B )
spcimedv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
Assertion
Ref Expression
spcimedv  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    B( x)

Proof of Theorem spcimedv
StepHypRef Expression
1 spcimedv.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
21ex 115 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ch  ->  ps ) ) )
32alrimiv 1885 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ch  ->  ps ) ) )
4 spcimdv.1 . 2  |-  ( ph  ->  A  e.  B )
5 nfv 1539 . . 3  |-  F/ x ch
6 nfcv 2332 . . 3  |-  F/_ x A
75, 6spcimegft 2830 . 2  |-  ( A. x ( x  =  A  ->  ( ch  ->  ps ) )  -> 
( A  e.  B  ->  ( ch  ->  E. x ps ) ) )
83, 4, 7sylc 62 1  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754
This theorem is referenced by:  rspcimedv  2858  fihashf1rn  10800
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