Theorem rabxmdc 3360

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 804	. . . . . 6 |- (DECID ph -> ( ph \/ -. ph ) )
2	1	a1d 22	. . . . 5 |- (DECID ph -> ( x e. A -> ( ph \/ -. ph ) ) )
3	2	alimi 1414	. . . 4 |- ( A. xDECID ph -> A. x ( x e. A -> ( ph \/ -. ph ) ) )
4		df-ral 2395	. . . 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 2581	. . 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 3313	. 2 |- ( { x e. A | ph } u. { x e. A | -. ph } ) = { x e. A | ( ph \/ -. ph ) }
9	7, 8	syl6eqr 2165	1 |- ( A. xDECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 680  DECID wdc 802   A.wal 1312   = wceq 1314   e. wcel 1463   A.wral 2390   {crab 2394   u. cun 3035

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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rab 2399  df-v 2659  df-un 3041

This theorem is referenced by: (None)