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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfsb7 2001* | An alternate definition of proper substitution df-sb 1773. By introducing a dummy variable 𝑧 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑥, 𝑦, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑧 effectively insulates 𝑥 from 𝑦. To achieve this, we use a chain of two substitutions in the form of sb5 1897, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 2002 provides a version where 𝜑 and 𝑧 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | ||
Theorem | sb7f 2002* | This version of dfsb7 2001 does not require that 𝜑 and 𝑧 be disjoint. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1536, i.e., that does not have the concept of a variable not occurring in a formula. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → ∀𝑧𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | ||
Theorem | sb7af 2003* | An alternate definition of proper substitution df-sb 1773. Similar to dfsb7a 2004 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 2002 in that it involves a dummy variable 𝑧, but expressed in terms of ∀ rather than ∃. (Contributed by Jim Kingdon, 5-Feb-2018.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | ||
Theorem | dfsb7a 2004* | An alternate definition of proper substitution df-sb 1773. Similar to dfsb7 2001 in that it involves a dummy variable 𝑧, but expressed in terms of ∀ rather than ∃. For a version which only requires Ⅎ𝑧𝜑 rather than 𝑧 and 𝜑 being distinct, see sb7af 2003. (Contributed by Jim Kingdon, 5-Feb-2018.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | ||
Theorem | sb10f 2005* | Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑)) | ||
Theorem | sbid2v 2006* | An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) | ||
Theorem | sbelx 2007* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑)) | ||
Theorem | sbel2x 2008* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑)) | ||
Theorem | sbalyz 2009* | Move universal quantifier in and out of substitution. Identical to sbal 2010 except that it has an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.) |
⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | ||
Theorem | sbal 2010* | Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | ||
Theorem | sbal1yz 2011* | Lemma for proving sbal1 2012. Same as sbal1 2012 but with an additional disjoint variable condition on 𝑦, 𝑧. (Contributed by Jim Kingdon, 23-Feb-2018.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | sbal1 2012* | A theorem used in elimination of disjoint variable conditions on 𝑥, 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | sbexyz 2013* | Move existential quantifier in and out of substitution. Identical to sbex 2014 except that it has an additional disjoint variable condition on 𝑦, 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.) |
⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) | ||
Theorem | sbex 2014* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) | ||
Theorem | sbalv 2015* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) | ||
Theorem | sbco4lem 2016* | Lemma for sbco4 2017. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.) |
⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | sbco4 2017* | Two ways of exchanging two variables. Both sides of the biconditional exchange 𝑥 and 𝑦, either via two temporary variables 𝑢 and 𝑣, or a single temporary 𝑤. (Contributed by Jim Kingdon, 25-Sep-2018.) |
⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | exsb 2018* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | 2exsb 2019* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) | ||
Theorem | dvelimALT 2020* | Version of dvelim 2027 that doesn't use ax-10 1515. Because it has different distinct variable constraints than dvelim 2027 and is used in important proofs, it would be better if it had a name which does not end in ALT (ideally more close to set.mm naming). (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | dvelimfv 2021* | Like dvelimf 2025 but with a distinct variable constraint on 𝑥 and 𝑧. (Contributed by Jim Kingdon, 6-Mar-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑧𝜓) & ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | hbsb4 2022 | A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
⊢ (𝜑 → ∀𝑧𝜑) ⇒ ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)) | ||
Theorem | hbsb4t 2023 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2022). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))) | ||
Theorem | nfsb4t 2024 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2022). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.) |
⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) | ||
Theorem | dvelimf 2025 | Version of dvelim 2027 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑧𝜓) & ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | dvelimdf 2026 | Deduction form of dvelimf 2025. This version may be useful if we want to avoid ax-17 1536 and use ax-16 1824 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑧𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑧𝜒) & ⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) | ||
Theorem | dvelim 2027* |
This theorem can be used to eliminate a distinct variable restriction on
𝑥 and 𝑧 and replace it with the
"distinctor" ¬ ∀𝑥𝑥 = 𝑦
as an antecedent. 𝜑 normally has 𝑧 free and can be read
𝜑(𝑧), and 𝜓 substitutes 𝑦 for
𝑧
and can be read
𝜑(𝑦). We don't require that 𝑥 and
𝑦
be distinct: if
they aren't, the distinctor will become false (in multiple-element
domains of discourse) and "protect" the consequent.
To obtain a closed-theorem form of this inference, prefix the hypotheses with ∀𝑥∀𝑧, conjoin them, and apply dvelimdf 2026. Other variants of this theorem are dvelimf 2025 (with no distinct variable restrictions) and dvelimALT 2020 (that avoids ax-10 1515). (Contributed by NM, 23-Nov-1994.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | dvelimor 2028* | Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula 𝜑 (containing 𝑧) and a distinct variable constraint between 𝑥 and 𝑧. The theorem makes it possible to replace the distinct variable constraint with the disjunct ∀𝑥𝑥 = 𝑦 (𝜓 is just a version of 𝜑 with 𝑦 substituted for 𝑧). (Contributed by Jim Kingdon, 11-May-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥𝜓) | ||
Theorem | dveeq1 2029* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
Theorem | sbal2 2030* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | nfsb4or 2031 | A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑) | ||
Theorem | nfd2 2032 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.) |
⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Theorem | hbe1a 2033 | Dual statement of hbe1 1505. (Contributed by Wolf Lammen, 15-Sep-2021.) |
⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | ||
Theorem | nf5-1 2034 | One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) | ||
Theorem | nf5d 2035 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Syntax | weu 2036 | Extend wff definition to include existential uniqueness ("there exists a unique 𝑥 such that 𝜑"). |
wff ∃!𝑥𝜑 | ||
Syntax | wmo 2037 | Extend wff definition to include uniqueness ("there exists at most one 𝑥 such that 𝜑"). |
wff ∃*𝑥𝜑 | ||
Theorem | eujust 2038* | A soundness justification theorem for df-eu 2039, showing that the definition is equivalent to itself with its dummy variable renamed. Note that 𝑦 and 𝑧 needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | ||
Definition | df-eu 2039* | Define existential uniqueness, i.e., "there exists exactly one 𝑥 such that 𝜑". Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2061, eu2 2080, eu3 2082, and eu5 2083 (which in some cases we show with a hypothesis 𝜑 → ∀𝑦𝜑 in place of a distinct variable condition on 𝑦 and 𝜑). Double uniqueness is tricky: ∃!𝑥∃!𝑦𝜑 does not mean "exactly one 𝑥 and one 𝑦 " (see 2eu4 2129). (Contributed by NM, 5-Aug-1993.) |
⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Definition | df-mo 2040 | Define "there exists at most one 𝑥 such that 𝜑". Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2090. For another possible definition see mo4 2097. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | ||
Theorem | euf 2041* | A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Theorem | eubidh 2042 | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | eubid 2043 | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | eubidv 2044* | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | eubii 2045 | Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓) | ||
Theorem | hbeu1 2046 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) |
⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) | ||
Theorem | nfeu1 2047 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥∃!𝑥𝜑 | ||
Theorem | nfmo1 2048 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥∃*𝑥𝜑 | ||
Theorem | sb8eu 2049 | Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8mo 2050 | Variable substitution for "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | nfeudv 2051* | Deduction version of nfeu 2055. Similar to nfeud 2052 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 25-May-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfeud 2052 | Deduction version of nfeu 2055. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfmod 2053 | Bound-variable hypothesis builder for "at most one". (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | ||
Theorem | nfeuv 2054* | Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2055 but has the additional condition that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 23-May-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦𝜑 | ||
Theorem | nfeu 2055 | Bound-variable hypothesis builder for existential uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦𝜑 | ||
Theorem | nfmo 2056 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦𝜑 | ||
Theorem | hbeu 2057 | Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑) | ||
Theorem | hbeud 2058 | Deduction version of hbeu 2057. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓)) | ||
Theorem | sb8euh 2059 | Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | cbveu 2060 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | eu1 2061* | An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))) | ||
Theorem | euor 2062 | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | euorv 2063* | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | mo2n 2064* | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | mon 2065 | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.) |
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | euex 2066 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | eumo0 2067* | Existential uniqueness implies "at most one". (Contributed by NM, 8-Jul-1994.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | eumo 2068 | Existential uniqueness implies "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | eumoi 2069 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
⊢ ∃!𝑥𝜑 ⇒ ⊢ ∃*𝑥𝜑 | ||
Theorem | mobidh 2070 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | mobid 2071 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | mobidv 2072* | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | mobii 2073 | Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) |
⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒) | ||
Theorem | hbmo1 2074 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.) |
⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑) | ||
Theorem | hbmo 2075 | Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑) | ||
Theorem | cbvmo 2076 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) | ||
Theorem | mo23 2077* | An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mor 2078* | Converse of mo23 2077 with an additional ∃𝑥𝜑 condition. (Contributed by Jim Kingdon, 25-Jun-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑥𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | modc 2079* | Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | eu2 2080* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | eu3h 2081* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | eu3 2082* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | eu5 2083 | Uniqueness in terms of "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | ||
Theorem | exmoeu2 2084 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) | ||
Theorem | moabs 2085 | Absorption of existence condition by "at most one". (Contributed by NM, 4-Nov-2002.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | ||
Theorem | exmodc 2086 | If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑)) | ||
Theorem | exmonim 2087 | There is at most one of something which does not exist. Unlike exmodc 2086 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.) |
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | mo2r 2088* | A condition which implies "at most one". (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) | ||
Theorem | mo3h 2089* | Alternate definition of "at most one". 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.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mo3 2090* | Alternate definition of "at most one". 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.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mo2dc 2091* | Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | euan 2092 | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) | ||
Theorem | euanv 2093* | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) | ||
Theorem | euor2 2094 | Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) | ||
Theorem | sbmo 2095* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑) | ||
Theorem | mo4f 2096* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | mo4 2097* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | eu4 2098* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | ||
Theorem | exmoeudc 2099 | Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.) |
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))) | ||
Theorem | moim 2100 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) |
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