Theorem rexnalim 2539

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 2534	. 2 |- ( E. x e. A -. ph <-> E. x ( x e. A /\ -. ph ) )
2		exanaliim 1700	. . 3 |- ( E. x ( x e. A /\ -. ph ) -> -. A. x ( x e. A -> ph ) )
3		df-ral 2533	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4	2, 3	sylnibr 688	. 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 1400   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is referenced by:  nnral  2540  ralexim  2542  rexanaliim  2656  iundif2ss  4073  ixp0  7003  omniwomnimkv  7497  alzdvds  12599  pc2dvds  13087  ballotfilem4  13219  isnsgrp  13698  umgr2edg1  16364  umgr2edgneu  16367  nninfsellemeq  16962