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Theorem spcimedv 2706
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 114 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ch  ->  ps ) ) )
32alrimiv 1803 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ch  ->  ps ) ) )
4 spcimdv.1 . 2  |-  ( ph  ->  A  e.  B )
5 nfv 1467 . . 3  |-  F/ x ch
6 nfcv 2229 . . 3  |-  F/_ x A
75, 6spcimegft 2698 . 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 103   A.wal 1288    = wceq 1290   E.wex 1427    e. wcel 1439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622
This theorem is referenced by:  rspcimedv  2725  fihashf1rn  10258
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