Theorem alexdc 1599

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1625. (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 1522	. . 3 |- F/ x A. xDECID ph
2		notnotbdc 858	. . . 4 |- (DECID ph -> ( ph <-> -. -. ph ) )
3	2	sps 1518	. . 3 |- ( A. xDECID ph -> ( ph <-> -. -. ph ) )
4	1, 3	albid 1595	. 2 |- ( A. xDECID ph -> ( A. x ph <-> A. x -. -. ph ) )
5		alnex 1476	. 2 |- ( A. x -. -. ph <-> -. E. x -. ph )
6	4, 5	syl6bb 195	1 |- ( A. xDECID ph -> ( A. x ph <-> -. E. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 820   A.wal 1330   E.wex 1469

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 1424  ax-gen 1426  ax-ie2 1471  ax-4 1488  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438

This theorem is referenced by: (None)