Theorem 19.33bdc 1521

Description: Converse of 19.33 1373 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 1520 (Contributed by Jim Kingdon, 23-Apr-2018.)

Assertion

Ref	Expression
19.33bdc	⊢ (DECID ∃𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))

Proof of Theorem 19.33bdc

Step	Hyp	Ref	Expression
1		ianordc 799	. 2 ⊢ (DECID ∃𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓)))
2		19.33b2 1520	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
3	1, 2	syl6bi 152	1 ⊢ (DECID ∃𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742  ∀wal 1241  ∃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

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

This theorem is referenced by: (None)