Theorem alexim 1659

Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1633. (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		pm2.24 622	. . . . 5 |- ( ph -> ( -. ph -> F. ) )
2	1	alimi 1469	. . . 4 |- ( A. x ph -> A. x ( -. ph -> F. ) )
3		exim 1613	. . . 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 1542	. . . 4 |- F/ x F.
6	5	19.9 1658	. . 3 |- ( E. x F. <-> F. )
7	4, 6	imbitrdi 161	. 2 |- ( A. x ph -> ( E. x -. ph -> F. ) )
8		dfnot 1382	. 2 |- ( -. E. x -. ph <-> ( E. x -. ph -> F. ) )
9	7, 8	sylibr 134	1 |- ( A. x ph -> -. E. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1362   F. wfal 1369   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475

This theorem is referenced by:  exnalim  1660  exists2  2142