Theorem ralexim 2489

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

Assertion

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

Proof of Theorem ralexim

Step	Hyp	Ref	Expression
1		rexnalim 2486	. 2 |- ( E. x e. A -. ph -> -. A. x e. A ph )
2	1	con2i 628	1 |- ( A. x e. A ph -> -. E. x e. A -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wral 2475   E.wrex 2476

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  df-ral 2480  df-rex 2481

This theorem is referenced by: (None)