Theorem rabxmdc 3528

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 844	. . . . . 6 |- (DECID ph -> ( ph \/ -. ph ) )
2	1	a1d 22	. . . . 5 |- (DECID ph -> ( x e. A -> ( ph \/ -. ph ) ) )
3	2	alimi 1504	. . . 4 |- ( A. xDECID ph -> A. x ( x e. A -> ( ph \/ -. ph ) ) )
4		df-ral 2516	. . . 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 2711	. . 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 3480	. 2 |- ( { x e. A | ph } u. { x e. A | -. ph } ) = { x e. A | ( ph \/ -. ph ) }
9	7, 8	eqtr4di 2282	1 |- ( A. xDECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 716  DECID wdc 842   A.wal 1396   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   u. cun 3199

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805  df-un 3205

This theorem is referenced by: (None)