Theorem dfrex2dc 2521

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 2514	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2	1	dcbii 845	. . 3 |- (DECID E. x e. A ph <-> DECID E. x ( x e. A /\ ph ) )
3		dfexdc 1547	. . 3 |- (DECID E. x ( x e. A /\ ph ) -> ( E. x ( x e. A /\ ph ) <-> -. A. x -. ( x e. A /\ ph ) ) )
4	2, 3	sylbi 121	. 2 |- (DECID E. x e. A ph -> ( E. x ( x e. A /\ ph ) <-> -. A. x -. ( x e. A /\ ph ) ) )
5		df-ral 2513	. . . 4 |- ( A. x e. A -. ph <-> A. x ( x e. A -> -. ph ) )
6		imnan 694	. . . . 5 |- ( ( x e. A -> -. ph ) <-> -. ( x e. A /\ ph ) )
7	6	albii 1516	. . . 4 |- ( A. x ( x e. A -> -. ph ) <-> A. x -. ( x e. A /\ ph ) )
8	5, 7	bitri 184	. . 3 |- ( A. x e. A -. ph <-> A. x -. ( x e. A /\ ph ) )
9	8	notbii 672	. 2 |- ( -. A. x e. A -. ph <-> -. A. x -. ( x e. A /\ ph ) )
10	4, 1, 9	3bitr4g 223	1 |- (DECID E. x e. A ph -> ( E. x e. A ph <-> -. A. x e. A -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 839   A.wal 1393   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-gen 1495  ax-ie2 1540

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-fal 1401  df-ral 2513  df-rex 2514

This theorem is referenced by:  dfrex2fin  7061  exmidomniim  7304