Theorem rabxmdc 3446

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 831	. . . . . 6 |- (DECID ph -> ( ph \/ -. ph ) )
2	1	a1d 22	. . . . 5 |- (DECID ph -> ( x e. A -> ( ph \/ -. ph ) ) )
3	2	alimi 1448	. . . 4 |- ( A. xDECID ph -> A. x ( x e. A -> ( ph \/ -. ph ) ) )
4		df-ral 2453	. . . 4 |- ( A. x e. A ( ph \/ -. ph ) <-> A. x ( x e. A -> ( ph \/ -. ph ) ) )
5	3, 4	sylibr 133	. . 3 |- ( A. xDECID ph -> A. x e. A ( ph \/ -. ph ) )
6		rabid2 2646	. . 3 |- ( A = { x e. A | ( ph \/ -. ph ) } <-> A. x e. A ( ph \/ -. ph ) )
7	5, 6	sylibr 133	. 2 |- ( A. xDECID ph -> A = { x e. A | ( ph \/ -. ph ) } )
8		unrab 3398	. 2 |- ( { x e. A | ph } u. { x e. A | -. ph } ) = { x e. A | ( ph \/ -. ph ) }
9	7, 8	eqtr4di 2221	1 |- ( A. xDECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 703  DECID wdc 829   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   u. cun 3119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-un 3125

This theorem is referenced by: (None)