Theorem decidi 15731

Description: Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

Assertion

Ref	Expression
decidi	|- ( A DECIDin B -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )

Proof of Theorem decidi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dcin 15730	. 2 |- ( A DECIDin B <-> A. x e. B DECID x e. A )
2		df-dc 837	. . . 4 |- (DECID x e. A <-> ( x e. A \/ -. x e. A ) )
3	2	ralbii 2512	. . 3 |- ( A. x e. B DECID x e. A <-> A. x e. B ( x e. A \/ -. x e. A ) )
4		eleq1 2268	. . . . 5 |- ( x = X -> ( x e. A <-> X e. A ) )
5	4	notbid 669	. . . . 5 |- ( x = X -> ( -. x e. A <-> -. X e. A ) )
6	4, 5	orbi12d 795	. . . 4 |- ( x = X -> ( ( x e. A \/ -. x e. A ) <-> ( X e. A \/ -. X e. A ) ) )
7	6	rspccv 2874	. . 3 |- ( A. x e. B ( x e. A \/ -. x e. A ) -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )
8	3, 7	sylbi 121	. 2 |- ( A. x e. B DECID x e. A -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )
9	1, 8	sylbi 121	1 |- ( A DECIDin B -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 710  DECID wdc 836   = wceq 1373   e. wcel 2176   A.wral 2484   DECID_in wdcin 15729

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-dcin 15730

This theorem is referenced by:  decidin  15733