Theorem exalim 1516

Description: One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1515. (Contributed by Jim Kingdon, 29-Jul-2018.)

Assertion

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

Proof of Theorem exalim

Step	Hyp	Ref	Expression
1		alnex 1513	. . 3 |- ( A. x -. ph <-> -. E. x ph )
2	1	biimpi 120	. 2 |- ( A. x -. ph -> -. E. x ph )
3	2	con2i 628	1 |- ( E. x ph -> -. A. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1362   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-ie2 1508

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

This theorem is referenced by:  n0rf  3464  ax9vsep  4157