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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | equsb1ALT 2601 | Alternate version of equsb1 2530. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦))) ⇒ ⊢ 𝜃 | ||
Theorem | sb6fALT 2602 | Alternate version of sb6f 2537. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (𝜃 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb5fALT 2603 | Alternate version of sb5f 2538. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (𝜃 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | nfsb4tALT 2604 | Alternate version of nfsb4t 2539. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) ⇒ ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃)) | ||
Theorem | nfsb4ALT 2605 | Alternate version of nfsb4 2540. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃) | ||
Theorem | sbi1ALT 2606 | Alternate version of sbi1 2076. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) & ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓)))) ⇒ ⊢ (𝜂 → (𝜃 → 𝜏)) | ||
Theorem | sbi2ALT 2607 | Alternate version of sbi2 2310. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) & ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓)))) ⇒ ⊢ ((𝜃 → 𝜏) → 𝜂) | ||
Theorem | sbimALT 2608 | Alternate version of sbim 2311. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) & ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓)))) ⇒ ⊢ (𝜂 ↔ (𝜃 → 𝜏)) | ||
Theorem | sbrimALT 2609 | Alternate version of sbrim 2313. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) & ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓)))) & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜂 ↔ (𝜑 → 𝜏)) | ||
Theorem | sbanALT 2610 | Alternate version of sban 2086. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) & ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ∧ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ∧ 𝜓)))) ⇒ ⊢ (𝜂 ↔ (𝜃 ∧ 𝜏)) | ||
Theorem | sbbiALT 2611 | Alternate version of sbbi 2317. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) & ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓)))) ⇒ ⊢ (𝜂 ↔ (𝜃 ↔ 𝜏)) | ||
Theorem | sblbisALT 2612 | Alternate version of sblbis 2319. (Contributed by NM, 19-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) & ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓)))) & ⊢ (𝜏 ↔ 𝜒) ⇒ ⊢ (𝜂 ↔ (𝜃 ↔ 𝜒)) | ||
Theorem | sbieALT 2613 | Alternate version of sbie 2544. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜃 ↔ 𝜓) | ||
Theorem | sbiedALT 2614 | Alternate version of sbied 2545. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (𝜏 ↔ 𝜒)) | ||
Theorem | sbco2ALT 2615 | Alternate version of sbco2 2553. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ (𝜏 ↔ ((𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) ∧ ∃𝑧(𝑧 = 𝑦 ∧ ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))))) & ⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (𝜏 ↔ 𝜃) | ||
Theorem | sb7fALT 2616* | Alternate version of sb7f 2568. (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) & ⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (𝜃 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | ||
Theorem | dfsb7ALT 2617* | Alternate version of dfsb7 2285. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) ⇒ ⊢ (𝜃 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | ||
Theorem | dfmoeu 2618* | An elementary proof of moeu 2668 in disguise, connecting an expression characterizing uniqueness (df-mo 2622) to that of existential uniqueness (eu6 2659). No particular order of definition is required, as one can be derived from the other. This is shown here and in dfeumo 2619. (Contributed by Wolf Lammen, 27-May-2019.) |
⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | dfeumo 2619* | An elementary proof showing the reverse direction of dfmoeu 2618. Here the characterizing expression of existential uniqueness (eu6 2659) is derived from that of uniqueness (df-mo 2622). (Contributed by Wolf Lammen, 3-Oct-2023.) |
⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Syntax | wmo 2620 | Extend wff definition to include the at-most-one quantifier ("there exists at most one 𝑥 such that 𝜑"). |
wff ∃*𝑥𝜑 | ||
Theorem | mojust 2621* | Soundness justification theorem for df-mo 2622 (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). (Contributed by NM, 11-Mar-2010.) Added this theorem by adapting the proof of eujust 2656. (Revised by BJ, 30-Sep-2022.) |
⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | ||
Definition | df-mo 2622* |
Define the at-most-one quantifier. The expression ∃*𝑥𝜑 is read
"there exists at most one 𝑥 such that 𝜑". This is also
called
the "uniqueness quantifier" but that expression is also used
for the
unique existential quantifier df-eu 2654, therefore we avoid that
ambiguous name.
Notation of [BellMachover] p. 460, whose definition we show as mo3 2648. For other possible definitions see moeu 2668 and mo4 2650. (Contributed by Wolf Lammen, 27-May-2019.) Make this the definition (which used to be moeu 2668, while this definition was then proved as dfmo 2682). (Revised by BJ, 30-Sep-2022.) |
⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | nexmo 2623 | Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2161. (Revised by Wolf Lammen, 16-Oct-2022.) |
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | exmo 2624 | Any proposition holds for some 𝑥 or holds for at most one 𝑥. (Contributed by NM, 8-Mar-1995.) Shorten proof and avoid df-eu 2654. (Revised by BJ, 14-Oct-2022.) |
⊢ (∃𝑥𝜑 ∨ ∃*𝑥𝜑) | ||
Theorem | moabs 2625 | Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2654. (Revised by BJ, 14-Oct-2022.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | ||
Theorem | moim 2626 | The at-most-one quantifier reverses implication. (Contributed by NM, 22-Apr-1995.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) | ||
Theorem | moimi 2627 | The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) Remove use of ax-5 1911. (Revised by Steven Nguyen, 9-May-2023.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑) | ||
Theorem | moimiOLD 2628 | Obsolete version of moimi 2627 as of 9-May-2023. The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑) | ||
Theorem | moimdv 2629* | The at-most-one quantifier reverses implication, deduction form. (Contributed by Thierry Arnoux, 25-Feb-2017.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓)) | ||
Theorem | mobi 2630 | Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)) | ||
Theorem | mobii 2631 | Formula-building rule for the at-most-one quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) Avoid ax-5 1911. (Revised by Wolf Lammen, 24-Sep-2023.) |
⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒) | ||
Theorem | mobiiOLD 2632 | Obsolete version of mobii 2631 as of 24-Sep-2023. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒) | ||
Theorem | mobidv 2633* | Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | mobid 2634 | Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2145, ax-11 2161, ax-13 2390. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | moa1 2635 | If an implication holds for at most one value, then its consequent holds for at most one value. See also ala1 1814 and exa1 1838. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (Revised by BJ, 29-Mar-2021.) |
⊢ (∃*𝑥(𝜑 → 𝜓) → ∃*𝑥𝜓) | ||
Theorem | moan 2636 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑)) | ||
Theorem | moani 2637 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
⊢ ∃*𝑥𝜑 ⇒ ⊢ ∃*𝑥(𝜓 ∧ 𝜑) | ||
Theorem | moor 2638 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑) | ||
Theorem | mooran1 2639 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | mooran2 2640 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓)) | ||
Theorem | nfmo1 2641 | Bound-variable hypothesis builder for the at-most-one quantifier. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) Adapt to new definition. (Revised by BJ, 1-Oct-2022.) |
⊢ Ⅎ𝑥∃*𝑥𝜑 | ||
Theorem | nfmod2 2642 | Bound-variable hypothesis builder for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2390. See nfmodv 2643 for a version replacing the distinctor with a disjoint variable condition, not requiring ax-13 2390. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid df-eu 2654. (Revised by BJ, 14-Oct-2022.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | ||
Theorem | nfmodv 2643* | Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2645 for a version without disjoint variable conditions but requiring ax-13 2390. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by BJ, 28-Jan-2023.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | ||
Theorem | nfmov 2644* | Bound-variable hypothesis builder for the at-most-one quantifier. See nfmo 2646 for a version without disjoint variable conditions but requiring ax-13 2390. (Contributed by NM, 9-Mar-1995.) (Revised by Wolf Lammen, 2-Oct-2023.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦𝜑 | ||
Theorem | nfmod 2645 | Bound-variable hypothesis builder for the at-most-one quantifier. Deduction version of nfmo 2646. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfmodv 2643 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | ||
Theorem | nfmo 2646 | Bound-variable hypothesis builder for the at-most-one quantifier. Note that 𝑥 and 𝑦 need not be disjoint. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfmov 2644 when possible. (Contributed by NM, 9-Mar-1995.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦𝜑 | ||
Theorem | mof 2647* | Version of df-mo 2622 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2682 from this proof, and prove mof 2647 from it (as of 30-Sep-2022, directly from df-mo 2622). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2390. (Revised by Wolf Lammen, 16-Oct-2022.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | mo3 2648* | Alternate definition of the at-most-one quantifier. Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) Remove dependency on ax-13 2390. (Revised by BJ and WL, 29-Jan-2023.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mo 2649* | Equivalent definitions of "there exists at most one". (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mo4 2650* |
At-most-one quantifier expressed using implicit substitution. This
theorem is also a direct consequence of mo4f 2651,
but this proof is based
on fewer axioms.
By the way, swapping 𝑥, 𝑦 and 𝜑, 𝜓 leads to an expression for ∃*𝑦𝜓, which is equivalent to ∃*𝑥𝜑 (is a proof line), so the right hand side is a rare instance of an expression where swapping the quantifiers can be done without ax-11 2161. (Contributed by NM, 26-Jul-1995.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2023.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | mo4f 2651* | At-most-one quantifier expressed using implicit substitution. Note that the disjoint variable condition on 𝑦, 𝜑 can be replaced by the nonfreeness hypothesis ⊢ Ⅎ𝑦𝜑 with essentially the same proof. (Contributed by NM, 10-Apr-2004.) Remove dependency on ax-13 2390. (Revised by Wolf Lammen, 19-Jan-2023.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | mo4OLD 2652* | Obsolete version of mo4 2650 as of 18-Oct-2023. (Contributed by NM, 26-Jul-1995.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Syntax | weu 2653 | Extend wff definition to include the unique existential quantifier ("there exists a unique 𝑥 such that 𝜑"). |
wff ∃!𝑥𝜑 | ||
Definition | df-eu 2654 |
Define the existential uniqueness quantifier. This expresses unique
existence, or existential uniqueness, which is the conjunction of
existence (df-ex 1781) and uniqueness (df-mo 2622). The expression
∃!𝑥𝜑 is read "there exists exactly
one 𝑥 such that 𝜑 " or
"there exists a unique 𝑥 such that 𝜑". This is also
called the
"uniqueness quantifier" but that expression is also used for the
at-most-one quantifier df-mo 2622, therefore we avoid that ambiguous name.
Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2694, eu2 2693, eu3v 2655, and eu6 2659. As for double unique existence, beware that the expression ∃!𝑥∃!𝑦𝜑 means "there exists a unique 𝑥 such that there exists a unique 𝑦 such that 𝜑 " which is a weaker property than "there exists exactly one 𝑥 and one 𝑦 such that 𝜑 " (see 2eu4 2739). (Contributed by NM, 12-Aug-1993.) Make this the definition (which used to be eu6 2659, while this definition was then proved as dfeu 2681). (Revised by BJ, 30-Sep-2022.) |
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | ||
Theorem | eu3v 2655* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Replace a nonfreeness hypothesis with a disjoint variable condition on 𝜑, 𝑦 to reduce axiom usage. (Revised by Wolf Lammen, 29-May-2019.) |
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | eujust 2656* | Soundness justification theorem for eu6 2659 when this was the definition of the unique existential quantifier (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). See eujustALT 2657 for a proof that provides an example of how it can be achieved through the use of dvelim 2473. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | ||
Theorem | eujustALT 2657* | Alternate proof of eujust 2656 illustrating the use of dvelim 2473. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | ||
Theorem | eu6lem 2658* | Lemma of eu6im 2660. A dissection of an idiom characterizing existential uniqueness. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2654 was then proved as dfeu 2681. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Extract common proof lines. (Revised by Wolf Lammen, 3-Mar-2023.) |
⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))) | ||
Theorem | eu6 2659* | Alternate definition of the unique existential quantifier df-eu 2654 not using the at-most-one quantifier. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2654 was then proved as dfeu 2681. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2161. (Revised by SN, 21-Sep-2023.) |
⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Theorem | eu6im 2660* | One direction of eu6 2659 needs fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.) |
⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃!𝑥𝜑) | ||
Theorem | euf 2661* | Version of eu6 2659 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) Avoid ax-13 2390. (Revised by Wolf Lammen, 16-Oct-2022.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Theorem | euex 2662 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.) |
⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | eumo 2663 | Existential uniqueness implies uniqueness. (Contributed by NM, 23-Mar-1995.) |
⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | eumoi 2664 | Uniqueness inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
⊢ ∃!𝑥𝜑 ⇒ ⊢ ∃*𝑥𝜑 | ||
Theorem | exmoeub 2665 | Existence implies that uniqueness is equivalent to unique existence. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) | ||
Theorem | exmoeu 2666 | Existence is equivalent to uniqueness implying existential uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.) |
⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)) | ||
Theorem | moeuex 2667 | Uniqueness implies that existence is equivalent to unique existence. (Contributed by BJ, 7-Oct-2022.) |
⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑)) | ||
Theorem | moeu 2668 | Uniqueness is equivalent to existence implying unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by NM, 8-Mar-1995.) This used to be the definition of the at-most-one quantifier, while df-mo 2622 was then proved as dfmo 2682. (Revised by BJ, 30-Sep-2022.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | ||
Theorem | eubi 2669 | Equivalence theorem for the unique existential quantifier. Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) Reduce dependencies on axioms. (Revised by BJ, 7-Oct-2022.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)) | ||
Theorem | eubii 2670 | Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) Avoid ax-5 1911. (Revised by Wolf Lammen, 27-Sep-2023.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓) | ||
Theorem | eubiiOLD 2671 | Obsolete version of eubii 2670 as of 27-Sep-2023. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓) | ||
Theorem | eubidv 2672* | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | eubid 2673 | Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 19-Feb-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | nfeu1 2674 | Bound-variable hypothesis builder for uniqueness. See also nfeu1ALT 2675. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥∃!𝑥𝜑 | ||
Theorem | nfeu1ALT 2675 | Alternate proof of nfeu1 2674. This illustrates the systematic way of proving nonfreeness in a defined expression: consider the definiens as a tree whose nodes are its subformulas, and prove by tree-induction nonfreeness of each node, starting from the leaves (generally using nfv 1915 or nf* theorems for previously defined expressions) and up to the root. Here, the definiens is a conjunction of two previously defined expressions, which automatically yields the present proof. (Contributed by BJ, 2-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥∃!𝑥𝜑 | ||
Theorem | nfeud2 2676 | Bound-variable hypothesis builder for uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2390. Check out nfeudw 2677 for a version that replaces the distinctor with a disjoint variable condition, not requiring ax-13 2390. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfeudw 2677* | Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2680. Version of nfeud 2678 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfeud 2678 | Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2680. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfeudw 2677 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfeuw 2679* | Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2680 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 8-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦𝜑 | ||
Theorem | nfeu 2680 | Bound-variable hypothesis builder for the unique existential quantifier. Note that 𝑥 and 𝑦 need not be disjoint. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfeuw 2679 when possible. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦𝜑 | ||
Theorem | dfeu 2681 | Rederive df-eu 2654 from the old definition eu6 2659. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) (Proof shortened by BJ, 7-Oct-2022.) (Proof modification is discouraged.) Use df-eu 2654 instead. (New usage is discouraged.) |
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | ||
Theorem | dfmo 2682* | Rederive df-mo 2622 from the old definition moeu 2668. (Contributed by Wolf Lammen, 27-May-2019.) (Proof modification is discouraged.) Use df-mo 2622 instead. (New usage is discouraged.) |
⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | euequ 2683* | There exists a unique set equal to a given set. Special case of eueqi 3702 proved using only predicate calculus. The proof needs 𝑦 = 𝑧 be free of 𝑥. This is ensured by having 𝑥 and 𝑦 be distinct. Alternately, a distinctor ¬ ∀𝑥𝑥 = 𝑦 could have been used instead. See eueq 3701 and eueqi 3702 for classes. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) Reduce axiom usage. (Revised by Wolf Lammen, 1-Mar-2023.) |
⊢ ∃!𝑥 𝑥 = 𝑦 | ||
Theorem | sb8eulem 2684* | Lemma. Factor out the common proof skeleton of sb8euv 2685 and sb8eu 2686. Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023.) |
⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8euv 2685* | Variable substitution in unique existential quantifier. Version of sb8eu 2686 requiring more disjoint variables, but fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Wolf Lammen, 7-Feb-2023.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8eu 2686 | Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2390. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2685. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8mo 2687 | Variable substitution for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | cbvmow 2688* | Rule used to change bound variables, using implicit substitution. Version of cbvmo 2689 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) | ||
Theorem | cbvmo 2689 | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbvmow 2688 when possible. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 4-Jan-2023.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) | ||
Theorem | cbveuw 2690* | Version of cbveu 2691 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | cbveu 2691 | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbveuw 2690 when possible. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | cbveuALT 2692 | Alternative proof of cbveu 2691. Since df-eu 2654 combines two other quantifiers, one can base this theorem on their associated 'change bounded variable' kind of theorems as well. (Contributed by Wolf Lammen, 5-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | eu2 2693* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | eu1 2694* | An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.) Avoid ax-13 2390. (Revised by Wolf Lammen, 7-Feb-2023.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))) | ||
Theorem | euor 2695 | Introduce a disjunct into a unique existential quantifier. For a version requiring disjoint variables, but fewer axioms, see euorv 2696. (Contributed by NM, 21-Oct-2005.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | euorv 2696* | Introduce a disjunct into a unique existential quantifier. Version of euor 2695 requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.) |
⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | euor2 2697 | Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) | ||
Theorem | sbmo 2698* | Substitution into an at-most-one quantifier. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑) | ||
Theorem | eu4 2699* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | ||
Theorem | euimmo 2700 | Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑)) |
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