Theorem a4sbe 1285

Description: A specialization theorem.

Assertion

Ref	Expression
a4sbe	|- ([y / x]ph -> E.xph)

Proof of Theorem a4sbe

Step	Hyp	Ref	Expression
1		stdpc4 1227	. . . 4 |- (A.x -. ph -> [y / x] -. ph)
2		sbn 1273	. . . 4 |- ([y / x] -. ph <-> -. [y / x]ph)
3	1, 2	sylib 205	. . 3 |- (A.x -. ph -> -. [y / x]ph)
4	3	con2i 102	. 2 |- ([y / x]ph -> -. A.x -. ph)
5		df-ex 1022	. 2 |- (E.xph <-> -. A.x -. ph)
6	4, 5	sylibr 207	1 |- ([y / x]ph -> E.xph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 995  E.wex 1021  [wsbc 1212

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-10 1007  ax-12 1009  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-11o 1260

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-sb 1214

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