Theorem sbea4 894

Description: A specialization theorem.

Assertion

Ref	Expression
sbea4	|- ([y / x]ph -> E.xph)

Proof of Theorem sbea4

Step	Hyp	Ref	Expression
1		stdpc4 869	. . . 4 |- (A.x -. ph -> [y / x] -. ph)
2		sbn 882	. . . 4 |- ([y / x] -. ph <-> -. [y / x]ph)
3	1, 2	sylib 173	. . 3 |- (A.x -. ph -> -. [y / x]ph)
4	3	con2i 89	. 2 |- ([y / x]ph -> -. A.x -. ph)
5		df-ex 679	. 2 |- (E.xph <-> -. A.x -. ph)
6	4, 5	sylibr 175	1 |- ([y / x]ph -> E.xph)

Syntax hints:  -. wn 1   -> wi 2  A.wal 672  E.wex 678  [wsb 852

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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