Theorem rexnalim 2359

Description: Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

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

Proof of Theorem rexnalim

Step	Hyp	Ref	Expression
1		df-rex 2354	. 2 |- ( E. x e. A -. ph <-> E. x ( x e. A /\ -. ph ) )
2		exanaliim 1578	. . 3 |- ( E. x ( x e. A /\ -. ph ) -> -. A. x ( x e. A -> ph ) )
3		df-ral 2353	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4	2, 3	sylnibr 634	. 2 |- ( E. x ( x e. A /\ -. ph ) -> -. A. x e. A ph )
5	1, 4	sylbi 119	1 |- ( E. x e. A -. ph -> -. A. x e. A ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421   e. wcel 1433   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-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by:  ralexim  2360  iundif2ss  3743  alzdvds  10254