Theorem rabxmdc 3454

Description: Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)

Assertion

Ref	Expression
rabxmdc	|- ( A. xDECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rabxmdc

Step	Hyp	Ref	Expression
1		exmiddc 836	. . . . . 6 |- (DECID ph -> ( ph \/ -. ph ) )
2	1	a1d 22	. . . . 5 |- (DECID ph -> ( x e. A -> ( ph \/ -. ph ) ) )
3	2	alimi 1455	. . . 4 |- ( A. xDECID ph -> A. x ( x e. A -> ( ph \/ -. ph ) ) )
4		df-ral 2460	. . . 4 |- ( A. x e. A ( ph \/ -. ph ) <-> A. x ( x e. A -> ( ph \/ -. ph ) ) )
5	3, 4	sylibr 134	. . 3 |- ( A. xDECID ph -> A. x e. A ( ph \/ -. ph ) )
6		rabid2 2653	. . 3 |- ( A = { x e. A | ( ph \/ -. ph ) } <-> A. x e. A ( ph \/ -. ph ) )
7	5, 6	sylibr 134	. 2 |- ( A. xDECID ph -> A = { x e. A | ( ph \/ -. ph ) } )
8		unrab 3406	. 2 |- ( { x e. A | ph } u. { x e. A | -. ph } ) = { x e. A | ( ph \/ -. ph ) }
9	7, 8	eqtr4di 2228	1 |- ( A. xDECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 708  DECID wdc 834   A.wal 1351   = wceq 1353   e. wcel 2148   A.wral 2455   {crab 2459   u. cun 3127

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2739  df-un 3133

This theorem is referenced by: (None)