Theorem rexalim 2459

Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

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

Proof of Theorem rexalim

Step	Hyp	Ref	Expression
1		ralnex 2454	. . 3 |- ( A. x e. A -. ph <-> -. E. x e. A ph )
2	1	biimpi 119	. 2 |- ( A. x e. A -. ph -> -. E. x e. A ph )
3	2	con2i 617	1 |- ( E. x e. A ph -> -. A. x e. A -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wral 2444   E.wrex 2445

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 1435  ax-gen 1437  ax-ie2 1482

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-ral 2449  df-rex 2450

This theorem is referenced by:  infnlbti  6991