Theorem alexim 1625

Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1599. (Contributed by Jim Kingdon, 2-Jul-2018.)

Assertion

Ref	Expression
alexim	|- ( A. x ph -> -. E. x -. ph )

Proof of Theorem alexim

Step	Hyp	Ref	Expression
1		pm2.24 611	. . . . 5 |- ( ph -> ( -. ph -> F. ) )
2	1	alimi 1432	. . . 4 |- ( A. x ph -> A. x ( -. ph -> F. ) )
3		exim 1579	. . . 4 |- ( A. x ( -. ph -> F. ) -> ( E. x -. ph -> E. x F. ) )
4	2, 3	syl 14	. . 3 |- ( A. x ph -> ( E. x -. ph -> E. x F. ) )
5		nfv 1509	. . . 4 |- F/ x F.
6	5	19.9 1624	. . 3 |- ( E. x F. <-> F. )
7	4, 6	syl6ib 160	. 2 |- ( A. x ph -> ( E. x -. ph -> F. ) )
8		dfnot 1350	. 2 |- ( -. E. x -. ph <-> ( E. x -. ph -> F. ) )
9	7, 8	sylibr 133	1 |- ( A. x ph -> -. E. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1330   F. wfal 1337   E.wex 1469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438

This theorem is referenced by:  exnalim  1626  exists2  2097