P. 21 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2001-2100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hbsbv 2001*	This is a version of hbsb 2009 with an extra distinct variable constraint, on 𝑧 and 𝑥. (Contributed by Jim Kingdon, 25-Dec-2017.)
⊢ (𝜑 → ∀𝑧𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Theorem	nfsbxy 2002*	Similar to hbsb 2009 but with an extra distinct variable constraint, on 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Mar-2018.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
Theorem	nfsbxyt 2003*	Closed form of nfsbxy 2002. (Contributed by Jim Kingdon, 9-May-2018.)
⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Theorem	sbco2vlem 2004*	This is a version of sbco2 2025 where 𝑧 is distinct from 𝑥 and from 𝑦. It is a lemma on the way to proving sbco2v 2008 which only requires that 𝑧 and 𝑥 be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) Remove one disjoint variable condition. (Revised by Jim Kingdon, 3-Feb-2018.)
⊢ (𝜑 → ∀𝑧𝜑)    ⇒   ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbco2vh 2005*	This is a version of sbco2 2025 where 𝑧 is distinct from 𝑥. (Contributed by Jim Kingdon, 12-Feb-2018.)
⊢ (𝜑 → ∀𝑧𝜑)    ⇒   ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	nfsb 2006*	If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
Theorem	nfsbv 2007*	If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is distinct from 𝑥 and 𝑦. Version of nfsb 2006 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
Theorem	sbco2v 2008*	Version of sbco2 2025 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	hbsb 2009*	If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
⊢ (𝜑 → ∀𝑧𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Theorem	equsb3lem 2010*	Lemma for equsb3 2011. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
Theorem	equsb3 2011*	Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
Theorem	sbn 2012	Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
Theorem	sbim 2013	Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sbor 2014	Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
Theorem	sban 2015	Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Theorem	sbrim 2016	Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sblim 2017	Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))
Theorem	sb3an 2018	Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓 ∧ [𝑦 / 𝑥]𝜒))
Theorem	sbbi 2019	Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Theorem	sblbis 2020	Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥](𝜒 ↔ 𝜑) ↔ ([𝑦 / 𝑥]𝜒 ↔ 𝜓))
Theorem	sbrbis 2021	Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ [𝑦 / 𝑥]𝜒))
Theorem	sbrbif 2022	Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
⊢ (𝜒 → ∀𝑥𝜒)    &   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))
Theorem	sbco2yz 2023*	This is a version of sbco2 2025 where 𝑧 is distinct from 𝑦. It is a lemma on the way to proving sbco2 2025 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbco2h 2024	A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
⊢ (𝜑 → ∀𝑧𝜑)    ⇒   ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbco2 2025	A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbco2d 2026	A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑧𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
Theorem	sbco2vd 2027*	Version of sbco2d 2026 with a distinct variable constraint between 𝑥 and 𝑧. (Contributed by Jim Kingdon, 19-Feb-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑧𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
Theorem	sbco 2028	A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbco3v 2029*	Version of sbco3 2034 with a distinct variable constraint between 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Feb-2018.)
⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
Theorem	sbcocom 2030	Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.)
⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)
Theorem	sbcomv 2031*	Version of sbcom 2035 with a distinct variable constraint between 𝑥 and 𝑧. (Contributed by Jim Kingdon, 28-Feb-2018.)
⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)
Theorem	sbcomxyyz 2032*	Version of sbcom 2035 with distinct variable constraints between 𝑥 and 𝑦, and 𝑦 and 𝑧. (Contributed by Jim Kingdon, 21-Mar-2018.)
⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)
Theorem	sbco3xzyz 2033*	Version of sbco3 2034 with distinct variable constraints between 𝑥 and 𝑧, and 𝑦 and 𝑧. Lemma for proving sbco3 2034. (Contributed by Jim Kingdon, 22-Mar-2018.)
⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
Theorem	sbco3 2034	A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
Theorem	sbcom 2035	A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)
Theorem	nfsbt 2036*	Closed form of nfsb 2006. (Contributed by Jim Kingdon, 9-May-2018.)
⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Theorem	nfsbd 2037*	Deduction version of nfsb 2006. (Contributed by NM, 15-Feb-2013.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑧𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
Theorem	sb9v 2038*	Like sb9 2039 but with a distinct variable constraint between 𝑥 and 𝑦. (Contributed by Jim Kingdon, 28-Feb-2018.)
⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb9 2039	Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb9i 2040	Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
⊢ (∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sbnf2 2041*	Two ways of expressing "𝑥 is (effectively) not free in 𝜑." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
Theorem	hbsbd 2042*	Deduction version of hbsb 2009. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑧𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓))
Theorem	2sb5 2043*	Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
Theorem	2sb6 2044*	Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
Theorem	sbcom2v 2045*	Lemma for proving sbcom2 2047. It is the same as sbcom2 2047 but with additional distinct variable constraints on 𝑥 and 𝑦, and on 𝑤 and 𝑧. (Contributed by Jim Kingdon, 19-Feb-2018.)
⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Theorem	sbcom2v2 2046*	Lemma for proving sbcom2 2047. It is the same as sbcom2v 2045 but removes the distinct variable constraint on 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Feb-2018.)
⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Theorem	sbcom2 2047*	Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.)
⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Theorem	sb6a 2048*	Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))
Theorem	2sb5rf 2049*	Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
⊢ (𝜑 → ∀𝑧𝜑)    &   ⊢ (𝜑 → ∀𝑤𝜑)    ⇒   ⊢ (𝜑 ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Theorem	2sb6rf 2050*	Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
⊢ (𝜑 → ∀𝑧𝜑)    &   ⊢ (𝜑 → ∀𝑤𝜑)    ⇒   ⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Theorem	dfsb7 2051*	An alternate definition of proper substitution df-sb 1816. 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 1942, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 2052 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 2052*	This version of dfsb7 2051 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 1579, 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 2053*	An alternate definition of proper substitution df-sb 1816. Similar to dfsb7a 2054 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 2052 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 2054*	An alternate definition of proper substitution df-sb 1816. Similar to dfsb7 2051 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 2053. (Contributed by Jim Kingdon, 5-Feb-2018.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
Theorem	sb10f 2055*	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 2056*	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 2057*	Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
Theorem	sbel2x 2058*	Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
Theorem	sbalyz 2059*	Move universal quantifier in and out of substitution. Identical to sbal 2060 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 2060*	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 2061*	Lemma for proving sbal1 2062. Same as sbal1 2062 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 2062*	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 2063*	Move existential quantifier in and out of substitution. Identical to sbex 2064 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 2064*	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 2065*	Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Theorem	sbco4lem 2066*	Lemma for sbco4 2067. 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 2067*	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 2068*	An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	2exsb 2069*	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 2070*	Version of dvelim 2077 that doesn't use ax-10 1558. Because it has different distinct variable constraints than dvelim 2077 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 2071*	Like dvelimf 2075 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 2072	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 2073	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2072). (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 2074	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2072). (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 2075	Version of dvelim 2077 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 2076	Deduction form of dvelimf 2075. This version may be useful if we want to avoid ax-17 1579 and use ax-16 1867 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 2077*	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 2076. Other variants of this theorem are dvelimf 2075 (with no distinct variable restrictions) and dvelimALT 2070 (that avoids ax-10 1558). (Contributed by NM, 23-Nov-1994.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Theorem	dvelimor 2078*	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 2079*	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 2080*	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 2081	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 2082	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 2083	Dual statement of hbe1 1548. (Contributed by Wolf Lammen, 15-Sep-2021.)
⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
Theorem	nf5-1 2084	One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
Theorem	nf5d 2085	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
1.4.6  Existential uniqueness
Syntax	weu 2086	Extend wff definition to include existential uniqueness ("there exists a unique 𝑥 such that 𝜑").
wff ∃!𝑥𝜑
Syntax	wmo 2087	Extend wff definition to include uniqueness ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑
Theorem	eujust 2088*	A soundness justification theorem for df-eu 2089, 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 2089*	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 2111, eu2 2131, eu3 2133, and eu5 2134 (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 2180). (Contributed by NM, 5-Aug-1993.)
⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
Definition	df-mo 2090	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 2141. For another possible definition see mo4 2148. (Contributed by NM, 5-Aug-1993.)
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
Theorem	euf 2091*	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 2092	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 2093	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Theorem	eubidv 2094*	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Theorem	eubii 2095	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 2096	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
Theorem	nfeu1 2097	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥∃!𝑥𝜑
Theorem	nfmo1 2098	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 2099	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 2100	Variable substitution for "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)