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Theorem alexdc 1492
Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1518. (Contributed by Jim Kingdon, 2-Jun-2018.)
Assertion
Ref Expression
alexdc (xDECID φ → (xφ ↔ ¬ x ¬ φ))

Proof of Theorem alexdc
StepHypRef Expression
1 nfa1 1416 . . 3 xxDECID φ
2 notnotdc 759 . . . 4 (DECID φ → (φ ↔ ¬ ¬ φ))
32sps 1412 . . 3 (xDECID φ → (φ ↔ ¬ ¬ φ))
41, 3albid 1488 . 2 (xDECID φ → (xφx ¬ ¬ φ))
5 alnex 1369 . 2 (x ¬ ¬ φ ↔ ¬ x ¬ φ)
64, 5syl6bb 185 1 (xDECID φ → (xφ ↔ ¬ x ¬ φ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  DECID wdc 733  wal 1226  wex 1362
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-gen 1318  ax-ie2 1364  ax-4 1381  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-dc 734  df-tru 1231  df-fal 1234  df-nf 1330
This theorem is referenced by: (None)
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