Theorem rabxmdc 3554

Description: Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Jim Kingdon, 17-Jun-2026.)

Assertion

Ref	Expression
rabxmdc	|- ( A. x e. A DECID 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 848	. . . 4 |- (DECID ph -> ( ph \/ -. ph ) )
2	1	ralimi 2613	. . 3 |- ( A. x e. A DECID ph -> A. x e. A ( ph \/ -. ph ) )
3		rabid2 2729	. . 3 |- ( A = { x e. A | ( ph \/ -. ph ) } <-> A. x e. A ( ph \/ -. ph ) )
4	2, 3	sylibr 134	. 2 |- ( A. x e. A DECID ph -> A = { x e. A | ( ph \/ -. ph ) } )
5		unrab 3504	. 2 |- ( { x e. A | ph } u. { x e. A | -. ph } ) = { x e. A | ( ph \/ -. ph ) }
6	4, 5	eqtr4di 2289	1 |- ( A. x e. A DECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 720  DECID wdc 846   = wceq 1402   A.wral 2528   {crab 2532   u. cun 3218

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-un 3224

This theorem is referenced by:  ballotfilemth  13259