Theorem decidi 15931

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 15930	. 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 2514	. . 3 |- ( A. x e. B DECID x e. A <-> A. x e. B ( x e. A \/ -. x e. A ) )
4		eleq1 2270	. . . . 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 2881	. . 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 2178   A.wral 2486   DECID_in wdcin 15929

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-dcin 15930

This theorem is referenced by:  decidin  15933