Theorem spsbe 2075

Description: A specialization theorem. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
spsbe	|- ([y / x]ph -> E.xph)

Proof of Theorem spsbe

Step	Hyp	Ref	Expression
1		stdpc4 2024	. . . 4 |- (A.x -. ph -> [y / x] -. ph)
2		sbn 2062	. . . 4 |- ([y / x] -. ph <-> -. [y / x]ph)
3	1, 2	sylib 188	. . 3 |- (A.x -. ph -> -. [y / x]ph)
4	3	con2i 112	. 2 |- ([y / x]ph -> -. A.x -. ph)
5		df-ex 1542	. 2 |- (E.xph <-> -. A.x -. ph)
6	4, 5	sylibr 203	1 |- ([y / x]ph -> E.xph)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)