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Theorem spcedv 2801
Description: Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.)
Hypotheses
Ref Expression
spcedv.1  |-  ( ph  ->  X  e.  _V )
spcedv.2  |-  ( ph  ->  ch )
spcedv.3  |-  ( x  =  X  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
spcedv  |-  ( ph  ->  E. x ps )
Distinct variable groups:    x, X    ch, x
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem spcedv
StepHypRef Expression
1 spcedv.1 . 2  |-  ( ph  ->  X  e.  _V )
2 spcedv.2 . 2  |-  ( ph  ->  ch )
3 spcedv.3 . . 3  |-  ( x  =  X  ->  ( ps 
<->  ch ) )
43spcegv 2800 . 2  |-  ( X  e.  _V  ->  ( ch  ->  E. x ps )
)
51, 2, 4sylc 62 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   _Vcvv 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714
This theorem is referenced by:  fprodseq  11480
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