Theorem decidr 12000

Description: Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

Hypothesis

Ref	Expression
decidr.1	|- ( ph -> ( x e. B -> ( x e. A \/ -. x e. A ) ) )

Assertion

Ref	Expression
decidr	|- ( ph -> A DECIDin B )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem decidr

Step	Hyp	Ref	Expression
1		decidr.1	. . . 4 |- ( ph -> ( x e. B -> ( x e. A \/ -. x e. A ) ) )
2		df-dc 782	. . . 4 |- (DECID x e. A <-> ( x e. A \/ -. x e. A ) )
3	1, 2	syl6ibr 161	. . 3 |- ( ph -> ( x e. B -> DECID x e. A ) )
4	3	alrimiv 1803	. 2 |- ( ph -> A. x ( x e. B -> DECID x e. A ) )
5		df-dcin 11998	. . 3 |- ( A DECIDin B <-> A. x e. B DECID x e. A )
6		df-ral 2365	. . 3 |- ( A. x e. B DECID x e. A <-> A. x ( x e. B -> DECID x e. A ) )
7	5, 6	bitri 183	. 2 |- ( A DECIDin B <-> A. x ( x e. B -> DECID x e. A ) )
8	4, 7	sylibr 133	1 |- ( ph -> A DECIDin B )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 665  DECID wdc 781   A.wal 1288   e. wcel 1439   A.wral 2360   DECID_in wdcin 11997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-17 1465

This theorem depends on definitions:  df-bi 116  df-dc 782  df-ral 2365  df-dcin 11998

This theorem is referenced by:  decidin  12001  uzdcinzz  12002  sumdc2  12003