Theorem alexdc 1510

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1536. (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 1434	. . 3 |- F/ x A. xDECID ph
2		notnotbdc 766	. . . 4 |- (DECID ph -> ( ph <-> -. -. ph ) )
3	2	sps 1430	. . 3 |- ( A. xDECID ph -> ( ph <-> -. -. ph ) )
4	1, 3	albid 1506	. 2 |- ( A. xDECID ph -> ( A. x ph <-> A. x -. -. ph ) )
5		alnex 1388	. 2 |- ( A. x -. -. ph <-> -. E. x -. ph )
6	4, 5	syl6bb 185	1 |- ( A. xDECID ph -> ( A. x ph <-> -. E. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98  DECID wdc 742   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249  df-nf 1350

This theorem is referenced by: (None)