Theorem rexnalim 2483

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 2478	. 2 |- ( E. x e. A -. ph <-> E. x ( x e. A /\ -. ph ) )
2		exanaliim 1658	. . 3 |- ( E. x ( x e. A /\ -. ph ) -> -. A. x ( x e. A -> ph ) )
3		df-ral 2477	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4	2, 3	sylnibr 678	. 2 |- ( E. x ( x e. A /\ -. ph ) -> -. A. x e. A ph )
5	1, 4	sylbi 121	1 |- ( E. x e. A -. ph -> -. A. x e. A ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1503   e. wcel 2164   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-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-ral 2477  df-rex 2478

This theorem is referenced by:  nnral  2484  ralexim  2486  iundif2ss  3978  ixp0  6785  omniwomnimkv  7226  alzdvds  11996  pc2dvds  12468  isnsgrp  12989  nninfsellemeq  15504