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Mirrors > Home > ILE Home > Th. List > 19.33bdc | GIF version |
Description: Converse of 19.33 1413 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 1560 (Contributed by Jim Kingdon, 23-Apr-2018.) |
Ref | Expression |
---|---|
19.33bdc | ⊢ (DECID ∃𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ianordc 832 | . 2 ⊢ (DECID ∃𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓))) | |
2 | 19.33b2 1560 | . 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))) | |
3 | 1, 2 | syl6bi 161 | 1 ⊢ (DECID ∃𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 DECID wdc 775 ∀wal 1282 ∃wex 1421 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-gen 1378 ax-ie2 1423 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-tru 1287 df-fal 1290 |
This theorem is referenced by: (None) |
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