Theorem alexdc 1550

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1576. (Contributed by Jim Kingdon, 2-Jun-2018.)

Assertion

Ref	Expression
alexdc	|- ( A. xDECID ph -> ( A. x ph <-> -. E. x -. ph ) )

Proof of Theorem alexdc

Step	Hyp	Ref	Expression
1		nfa1 1474	. . 3 |- F/ x A. xDECID ph
2		notnotbdc 799	. . . 4 |- (DECID ph -> ( ph <-> -. -. ph ) )
3	2	sps 1470	. . 3 |- ( A. xDECID ph -> ( ph <-> -. -. ph ) )
4	1, 3	albid 1546	. 2 |- ( A. xDECID ph -> ( A. x ph <-> A. x -. -. ph ) )
5		alnex 1428	. 2 |- ( A. x -. -. ph <-> -. E. x -. ph )
6	4, 5	syl6bb 194	1 |- ( A. xDECID ph -> ( A. x ph <-> -. E. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103  DECID wdc 775   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-fal 1290  df-nf 1390

This theorem is referenced by: (None)