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Theorem spsbe 2015
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 1964 . . . 4  |-  ( A. x  -.  ph  ->  [ y  /  x ]  -.  ph )
2 sbn 2002 . . . 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 1529 . 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 1527   E.wex 1528   [wsb 1629
This theorem is referenced by:  spsbce-2  27579  sb5ALT  28288  sb5ALTVD  28689
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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