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Theorem 19.33bdc 1592
Description: Converse of 19.33 1443 given ¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version which does not require a decidability condition, see 19.33b2 1591 (Contributed by Jim Kingdon, 23-Apr-2018.)
Assertion
Ref Expression
19.33bdc (DECID𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))

Proof of Theorem 19.33bdc
StepHypRef Expression
1 ianordc 867 . 2 (DECID𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓)))
2 19.33b2 1591 . 2 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
31, 2syl6bi 162 1 (DECID𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  DECID wdc 802  wal 1312  wex 1451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-gen 1408  ax-ie2 1453
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-fal 1320
This theorem is referenced by: (None)
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