Theorem rexalim 2487

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 2482	. . 3 |- ( A. x e. A -. ph <-> -. E. x e. A ph )
2	1	biimpi 120	. 2 |- ( A. x e. A -. ph -> -. E. x e. A ph )
3	2	con2i 628	1 |- ( E. x e. A ph -> -. A. x e. A -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wral 2472   E.wrex 2473

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 1458  ax-gen 1460  ax-ie2 1505

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-ral 2477  df-rex 2478

This theorem is referenced by:  infnlbti  7075