Theorem alexdc 1665

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1691. (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 1587	. . 3 |- F/ x A. xDECID ph
2		notnotbdc 877	. . . 4 |- (DECID ph -> ( ph <-> -. -. ph ) )
3	2	sps 1583	. . 3 |- ( A. xDECID ph -> ( ph <-> -. -. ph ) )
4	1, 3	albid 1661	. 2 |- ( A. xDECID ph -> ( A. x ph <-> A. x -. -. ph ) )
5		alnex 1545	. 2 |- ( A. x -. -. ph <-> -. E. x -. ph )
6	4, 5	bitrdi 196	1 |- ( A. xDECID ph -> ( A. x ph <-> -. E. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105  DECID wdc 839   A.wal 1393   E.wex 1538

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-gen 1495  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-fal 1401  df-nf 1507

This theorem is referenced by: (None)