Theorem decidi 13002

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 13001	. 2 |- ( A DECIDin B <-> A. x e. B DECID x e. A )
2		df-dc 820	. . . 4 |- (DECID x e. A <-> ( x e. A \/ -. x e. A ) )
3	2	ralbii 2441	. . 3 |- ( A. x e. B DECID x e. A <-> A. x e. B ( x e. A \/ -. x e. A ) )
4		eleq1 2202	. . . . 5 |- ( x = X -> ( x e. A <-> X e. A ) )
5	4	notbid 656	. . . . 5 |- ( x = X -> ( -. x e. A <-> -. X e. A ) )
6	4, 5	orbi12d 782	. . . 4 |- ( x = X -> ( ( x e. A \/ -. x e. A ) <-> ( X e. A \/ -. X e. A ) ) )
7	6	rspccv 2786	. . 3 |- ( A. x e. B ( x e. A \/ -. x e. A ) -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )
8	3, 7	sylbi 120	. 2 |- ( A. x e. B DECID x e. A -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )
9	1, 8	sylbi 120	1 |- ( A DECIDin B -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   A.wral 2416   DECID_in wdcin 13000

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-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dcin 13001

This theorem is referenced by:  decidin  13004