Theorem rexnalim 2455

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 2450	. 2 |- ( E. x e. A -. ph <-> E. x ( x e. A /\ -. ph ) )
2		exanaliim 1635	. . 3 |- ( E. x ( x e. A /\ -. ph ) -> -. A. x ( x e. A -> ph ) )
3		df-ral 2449	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4	2, 3	sylnibr 667	. 2 |- ( E. x ( x e. A /\ -. ph ) -> -. A. x e. A ph )
5	1, 4	sylbi 120	1 |- ( E. x e. A -. ph -> -. A. x e. A ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480   e. wcel 2136   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-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  nnral  2456  ralexim  2458  iundif2ss  3931  ixp0  6697  omniwomnimkv  7131  alzdvds  11792  pc2dvds  12261  nninfsellemeq  13894