Theorem decidr 14117

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 835	. . . 4 |- (DECID x e. A <-> ( x e. A \/ -. x e. A ) )
3	1, 2	syl6ibr 162	. . 3 |- ( ph -> ( x e. B -> DECID x e. A ) )
4	3	alrimiv 1872	. 2 |- ( ph -> A. x ( x e. B -> DECID x e. A ) )
5		df-dcin 14115	. . 3 |- ( A DECIDin B <-> A. x e. B DECID x e. A )
6		df-ral 2458	. . 3 |- ( A. x e. B DECID x e. A <-> A. x ( x e. B -> DECID x e. A ) )
7	5, 6	bitri 184	. 2 |- ( A DECIDin B <-> A. x ( x e. B -> DECID x e. A ) )
8	4, 7	sylibr 134	1 |- ( ph -> A DECIDin B )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 708  DECID wdc 834   A.wal 1351   e. wcel 2146   A.wral 2453   DECID_in wdcin 14114

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-5 1445  ax-gen 1447  ax-17 1524

This theorem depends on definitions:  df-bi 117  df-dc 835  df-ral 2458  df-dcin 14115

This theorem is referenced by:  decidin  14118  uzdcinzz  14119  sumdc2  14120