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Theorem spsbe 2028
Description: A specialization theorem. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
spsbe  |-  ( [ y  /  x ] ph  ->  E. x ph )

Proof of Theorem spsbe
StepHypRef Expression
1 stdpc4 1977 . . . 4  |-  ( A. x  -.  ph  ->  [ y  /  x ]  -.  ph )
2 sbn 2015 . . . 4  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
31, 2sylib 188 . . 3  |-  ( A. x  -.  ph  ->  -.  [
y  /  x ] ph )
43con2i 112 . 2  |-  ( [ y  /  x ] ph  ->  -.  A. x  -.  ph )
5 df-ex 1532 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
64, 5sylibr 203 1  |-  ( [ y  /  x ] ph  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   E.wex 1531   [wsb 1638
This theorem is referenced by:  spsbce-2  27682  sb5ALT  28587  sb5ALTVD  29005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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