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

Step	Hyp	Ref	Expression
1		stdpc4 2073	. . . 4 |- ( A. x -. ph -> [ y / x ] -. ph )
2		sbn 2111	. . . 4 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
3	1, 2	sylib 189	. . 3 |- ( A. x -. ph -> -. [ y / x ] ph )
4	3	con2i 114	. 2 |- ( [ y / x ] ph -> -. A. x -. ph )
5		df-ex 1548	. 2 |- ( E. x ph <-> -. A. x -. ph )
6	4, 5	sylibr 204	1 |- ( [ y / x ] ph -> E. x ph )

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