Theorem rexnalim 2522

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 2517	. 2 |- ( E. x e. A -. ph <-> E. x ( x e. A /\ -. ph ) )
2		exanaliim 1696	. . 3 |- ( E. x ( x e. A /\ -. ph ) -> -. A. x ( x e. A -> ph ) )
3		df-ral 2516	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4	2, 3	sylnibr 684	. 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 1396   E.wex 1541   e. wcel 2202   A.wral 2511   E.wrex 2512

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 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-ral 2516  df-rex 2517

This theorem is referenced by:  nnral  2523  ralexim  2525  rexanaliim  2639  iundif2ss  4041  ixp0  6943  omniwomnimkv  7426  alzdvds  12495  pc2dvds  12983  isnsgrp  13569  umgr2edg1  16150  umgr2edgneu  16153  nninfsellemeq  16740