Theorem decidi 16806

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 16805	. 2 |- ( A DECIDin B <-> A. x e. B DECID x e. A )
2		df-dc 847	. . . 4 |- (DECID x e. A <-> ( x e. A \/ -. x e. A ) )
3	2	ralbii 2556	. . 3 |- ( A. x e. B DECID x e. A <-> A. x e. B ( x e. A \/ -. x e. A ) )
4		eleq1 2301	. . . . 5 |- ( x = X -> ( x e. A <-> X e. A ) )
5	4	notbid 677	. . . . 5 |- ( x = X -> ( -. x e. A <-> -. X e. A ) )
6	4, 5	orbi12d 805	. . . 4 |- ( x = X -> ( ( x e. A \/ -. x e. A ) <-> ( X e. A \/ -. X e. A ) ) )
7	6	rspccv 2926	. . 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 720  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   DECID_in wdcin 16804

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dcin 16805

This theorem is referenced by:  decidin  16808