Theorem rexalim 2361

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 2358	. . 3 |- ( A. x e. A -. ph <-> -. E. x e. A ph )
2	1	biimpi 118	. 2 |- ( A. x e. A -. ph -> -. E. x e. A ph )
3	2	con2i 589	1 |- ( E. x e. A ph -> -. A. x e. A -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wral 2348   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie2 1423

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-ral 2353  df-rex 2354

This theorem is referenced by:  infnlbti  6439