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Theorem spsbe 2124
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 2073 . . . 4  |-  ( A. x  -.  ph  ->  [ y  /  x ]  -.  ph )
2 sbn 2111 . . . 4  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
31, 2sylib 189 . . 3  |-  ( A. x  -.  ph  ->  -.  [
y  /  x ] ph )
43con2i 114 . 2  |-  ( [ y  /  x ] ph  ->  -.  A. x  -.  ph )
5 df-ex 1548 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
64, 5sylibr 204 1  |-  ( [ y  /  x ] ph  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546   E.wex 1547   [wsb 1655
This theorem is referenced by:  spsbce-2  27447  sb5ALT  28320  sb5ALTVD  28734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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