Theorem dfrex2dc 2428

Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)

Assertion

Ref	Expression
dfrex2dc	|- (DECID E. x e. A ph -> ( E. x e. A ph <-> -. A. x e. A -. ph ) )

Proof of Theorem dfrex2dc

Step	Hyp	Ref	Expression
1		df-rex 2422	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2	1	dcbii 825	. . 3 |- (DECID E. x e. A ph <-> DECID E. x ( x e. A /\ ph ) )
3		dfexdc 1477	. . 3 |- (DECID E. x ( x e. A /\ ph ) -> ( E. x ( x e. A /\ ph ) <-> -. A. x -. ( x e. A /\ ph ) ) )
4	2, 3	sylbi 120	. 2 |- (DECID E. x e. A ph -> ( E. x ( x e. A /\ ph ) <-> -. A. x -. ( x e. A /\ ph ) ) )
5		df-ral 2421	. . . 4 |- ( A. x e. A -. ph <-> A. x ( x e. A -> -. ph ) )
6		imnan 679	. . . . 5 |- ( ( x e. A -> -. ph ) <-> -. ( x e. A /\ ph ) )
7	6	albii 1446	. . . 4 |- ( A. x ( x e. A -> -. ph ) <-> A. x -. ( x e. A /\ ph ) )
8	5, 7	bitri 183	. . 3 |- ( A. x e. A -. ph <-> A. x -. ( x e. A /\ ph ) )
9	8	notbii 657	. 2 |- ( -. A. x e. A -. ph <-> -. A. x -. ( x e. A /\ ph ) )
10	4, 1, 9	3bitr4g 222	1 |- (DECID E. x e. A ph -> ( E. x e. A ph <-> -. A. x e. A -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 819   A.wal 1329   E.wex 1468   e. wcel 1480   A.wral 2416   E.wrex 2417

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie2 1470

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-ral 2421  df-rex 2422

This theorem is referenced by:  dfrex2fin  6797  exmidomniim  7013