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Theorem 19.33bdc 1564
Description: Converse of 19.33 1416 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 1563 (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 835 . 2 (DECID𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓)))
2 19.33b2 1563 . 2 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
31, 2syl6bi 161 1 (DECID𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 778  wal 1285  wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-gen 1381  ax-ie2 1426
This theorem depends on definitions:  df-bi 115  df-dc 779  df-tru 1290  df-fal 1293
This theorem is referenced by: (None)
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