Theorem alexdc 1607

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1633. (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 1529	. . 3 |- F/ x A. xDECID ph
2		notnotbdc 862	. . . 4 |- (DECID ph -> ( ph <-> -. -. ph ) )
3	2	sps 1525	. . 3 |- ( A. xDECID ph -> ( ph <-> -. -. ph ) )
4	1, 3	albid 1603	. 2 |- ( A. xDECID ph -> ( A. x ph <-> A. x -. -. ph ) )
5		alnex 1487	. 2 |- ( A. x -. -. ph <-> -. E. x -. ph )
6	4, 5	bitrdi 195	1 |- ( A. xDECID ph -> ( A. x ph <-> -. E. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 824   A.wal 1341   E.wex 1480

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-gen 1437  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449

This theorem is referenced by: (None)