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Theorem List for Intuitionistic Logic Explorer - 1701-1800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Axiomax-11o 1701 Axiom ax-11o 1701 ("o" for "old") was the original version of ax-11 1394, before it was discovered (in Jan. 2007) that the shorter ax-11 1394 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of "¬ xx = y..." as informally meaning "if x and y are distinct variables then..." The antecedent becomes false if the same variable is substituted for x and y, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form ¬ xx = y a "distinctor."

This axiom is redundant, as shown by theorem ax11o 1700.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1700. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

x x = y → (x = y → (φx(x = yφ))))

1.4.3  More theorems related to ax-11 and substitution

Theoremalbidv 1702* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))       (φ → (xψxχ))

Theoremexbidv 1703* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))       (φ → (xψxχ))

Theoremax11b 1704 A bidirectional version of ax-11o 1701. (Contributed by NM, 30-Jun-2006.)
((¬ x x = y x = y) → (φx(x = yφ)))

Theoremax11v 1705* This is a version of ax-11o 1701 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)
(x = y → (φx(x = yφ)))

Theoremax11ev 1706* Analogue to ax11v 1705 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
(x = y → (x(x = y φ) → φ))

Theoremequs5 1707 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
x x = y → (x(x = y φ) → x(x = yφ)))

Theoremequs5or 1708 Lemma used in proofs of substitution properties. Like equs5 1707 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)
(x x = y (x(x = y φ) → x(x = yφ)))

Theoremsb3 1709 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
x x = y → (x(x = y φ) → [y / x]φ))

Theoremsb4 1710 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
x x = y → ([y / x]φx(x = yφ)))

Theoremsb4or 1711 One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1710 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
(x x = y x([y / x]φx(x = yφ)))

Theoremsb4b 1712 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
x x = y → ([y / x]φx(x = yφ)))

Theoremsb4bor 1713 Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
(x x = y x([y / x]φx(x = yφ)))

Theoremhbsb2 1714 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
x x = y → ([y / x]φx[y / x]φ))

Theoremnfsb2or 1715 Bound-variable hypothesis builder for substitution. Similar to hbsb2 1714 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)
(x x = y x[y / x]φ)

Theoremsbequilem 1716 Propositional logic lemma used in the sbequi 1717 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
(φ (ψ → (χθ)))    &   (τ (ψ → (θη)))       (φ (τ (ψ → (χη))))

Theoremsbequi 1717 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
(x = y → ([x / z]φ → [y / z]φ))

Theoremsbequ 1718 An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
(x = y → ([x / z]φ ↔ [y / z]φ))

Theoremdrsb2 1719 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
(x x = y → ([x / z]φ ↔ [y / z]φ))

Theoremspsbe 1720 A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)
([y / x]φxφ)

Theoremspsbim 1721 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
(x(φψ) → ([y / x]φ → [y / x]ψ))

Theoremspsbbi 1722 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
(x(φψ) → ([y / x]φ ↔ [y / x]ψ))

Theoremsbbid 1723 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φ → (ψχ))       (φ → ([y / x]ψ ↔ [y / x]χ))

Theoremsbequ8 1724 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.)
([y / x]φ ↔ [y / x](x = yφ))

Theoremsbft 1725 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
(Ⅎxφ → ([y / x]φφ))

Theoremsbid2h 1726 An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
(φxφ)       ([y / x][x / y]φφ)

Theoremsbid2 1727 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
xφ       ([y / x][x / y]φφ)

Theoremsbidm 1728 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
([y / x][y / x]φ ↔ [y / x]φ)

Theoremsb5rf 1729 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φyφ)       (φy(y = x [y / x]φ))

Theoremsb6rf 1730 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φyφ)       (φy(y = x → [y / x]φ))

Theoremsb8h 1731 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
(φyφ)       (xφy[y / x]φ)

Theoremsb8eh 1732 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
(φyφ)       (xφy[y / x]φ)

Theoremsb8 1733 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
yφ       (xφy[y / x]φ)

Theoremsb8e 1734 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
yφ       (xφy[y / x]φ)

1.4.4  Predicate calculus with distinct variables (cont.)

Theoremax16i 1735* Inference with ax-16 1692 as its conclusion, that doesn't require ax-10 1393, ax-11 1394, or ax-12 1399 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)
(x = z → (φψ))    &   (ψxψ)       (x x = y → (φxφ))

Theoremax16ALT 1736* Version of ax16 1691 that doesn't require ax-10 1393 or ax-12 1399 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(x x = y → (φxφ))

Theoremspv 1737* Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
(x = y → (φψ))       (xφψ)

Theoremspimev 1738* Distinct-variable version of spime 1626. (Contributed by NM, 5-Aug-1993.)
(x = y → (φψ))       (φxψ)

Theoremspeiv 1739* Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
(x = y → (φψ))    &   ψ       xφ

Theoremequvin 1740* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
(x = yz(x = z z = y))

Theorema16g 1741* A generalization of axiom ax-16 1692. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(x x = y → (φzφ))

Theorema16gb 1742* A generalization of axiom ax-16 1692. (Contributed by NM, 5-Aug-1993.)
(x x = y → (φzφ))

Theorema16nf 1743* If there is only one element in the universe, then everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.)
(x x = y → Ⅎzφ)

Theorem2albidv 1744* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
(φ → (ψχ))       (φ → (xyψxyχ))

Theorem2exbidv 1745* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(φ → (ψχ))       (φ → (xyψxyχ))

Theorem3exbidv 1746* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(φ → (ψχ))       (φ → (xyzψxyzχ))

Theorem4exbidv 1747* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
(φ → (ψχ))       (φ → (xyzwψxyzwχ))

Theorem19.9v 1748* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
(xφφ)

Theoremexlimdd 1749 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
xφ    &   xχ    &   (φxψ)    &   ((φ ψ) → χ)       (φχ)

Theorem19.21v 1750* Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as (φxφ) in 19.21 1472 via the use of distinct variable conditions combined with ax-17 1416. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 1902 derived from df-eu 1900. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.)
(x(φψ) ↔ (φxψ))

Theoremalrimiv 1751* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(φψ)       (φxψ)

Theoremalrimivv 1752* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
(φψ)       (φxyψ)

Theoremalrimdv 1753* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
(φ → (ψχ))       (φ → (ψxχ))

Theoremnfdv 1754* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(φ → (ψxψ))       (φ → Ⅎxψ)

Theorem2ax17 1755* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(φxyφ)

Theoremalimdv 1756* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
(φ → (ψχ))       (φ → (xψxχ))

Theoremeximdv 1757* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
(φ → (ψχ))       (φ → (xψxχ))

Theorem2alimdv 1758* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
(φ → (ψχ))       (φ → (xyψxyχ))

Theorem2eximdv 1759* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
(φ → (ψχ))       (φ → (xyψxyχ))

Theorem19.23v 1760* Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
(x(φψ) ↔ (xφψ))

Theorem19.23vv 1761* Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
(xy(φψ) ↔ (xyφψ))

Theoremsb56 1762* Two equivalent ways of expressing the proper substitution of y for x in φ, when x and y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1643. (Contributed by NM, 14-Apr-2008.)
(x(x = y φ) ↔ x(x = yφ))

Theoremsb6 1763* Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
([y / x]φx(x = yφ))

Theoremsb5 1764* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
([y / x]φx(x = y φ))

Theoremsbnv 1765* Version of sbn 1823 where x and y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
([y / x] ¬ φ ↔ ¬ [y / x]φ)

Theoremsbanv 1766* Version of sban 1826 where x and y are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))

Theoremsborv 1767* Version of sbor 1825 where x and y are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))

Theoremsbi1v 1768* Forward direction of sbimv 1770. (Contributed by Jim Kingdon, 25-Dec-2017.)
([y / x](φψ) → ([y / x]φ → [y / x]ψ))

Theoremsbi2v 1769* Reverse direction of sbimv 1770. (Contributed by Jim Kingdon, 18-Jan-2018.)
(([y / x]φ → [y / x]ψ) → [y / x](φψ))

Theoremsbimv 1770* Intuitionistic proof of sbim 1824 where x and y are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)
([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))

Theoremsblimv 1771* Version of sblim 1828 where x and y are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
(ψxψ)       ([y / x](φψ) ↔ ([y / x]φψ))

Theorempm11.53 1772* Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(xy(φψ) ↔ (xφyψ))

Theoremexlimivv 1773* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
(φψ)       (xyφψ)

Theoremexlimdvv 1774* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
(φ → (ψχ))       (φ → (xyψχ))

Theoremexlimddv 1775* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(φxψ)    &   ((φ ψ) → χ)       (φχ)

Theorem19.27v 1776* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
(x(φ ψ) ↔ (xφ ψ))

Theorem19.28v 1777* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
(x(φ ψ) ↔ (φ xψ))

Theorem19.36aiv 1778* Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
x(φψ)       (xφψ)

Theorem19.41v 1779* Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(x(φ ψ) ↔ (xφ ψ))

Theorem19.41vv 1780* Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
(xy(φ ψ) ↔ (xyφ ψ))

Theorem19.41vvv 1781* Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
(xyz(φ ψ) ↔ (xyzφ ψ))

Theorem19.41vvvv 1782* Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
(wxyz(φ ψ) ↔ (wxyzφ ψ))

Theorem19.42v 1783* Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(x(φ ψ) ↔ (φ xψ))

Theoremexdistr 1784* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(xy(φ ψ) ↔ x(φ yψ))

Theorem19.42vv 1785* Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)
(xy(φ ψ) ↔ (φ xyψ))

Theorem19.42vvv 1786* Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
(xyz(φ ψ) ↔ (φ xyzψ))

Theorem19.42vvvv 1787* Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
(wxyz(φ ψ) ↔ (φ wxyzψ))

Theoremexdistr2 1788* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(xyz(φ ψ) ↔ x(φ yzψ))

Theorem3exdistr 1789* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(xyz(φ ψ χ) ↔ x(φ y(ψ zχ)))

Theorem4exdistr 1790* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(xyzw((φ ψ) (χ θ)) ↔ x(φ y(ψ z(χ wθ))))

Theoremcbvalv 1791* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
(x = y → (φψ))       (xφyψ)

Theoremcbvexv 1792* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
(x = y → (φψ))       (xφyψ)

Theoremcbval2 1793* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
zφ    &   wφ    &   xψ    &   yψ    &   ((x = z y = w) → (φψ))       (xyφzwψ)

Theoremcbvex2 1794* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
zφ    &   wφ    &   xψ    &   yψ    &   ((x = z y = w) → (φψ))       (xyφzwψ)

Theoremcbval2v 1795* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
((x = z y = w) → (φψ))       (xyφzwψ)

Theoremcbvex2v 1796* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
((x = z y = w) → (φψ))       (xyφzwψ)

Theoremcbvald 1797* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1890. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
yφ    &   (φ → Ⅎyψ)    &   (φ → (x = y → (ψχ)))       (φ → (xψyχ))

Theoremcbvexdh 1798* Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1890. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
(φyφ)    &   (φ → (ψyψ))    &   (φ → (x = y → (ψχ)))       (φ → (xψyχ))

Theoremcbvexd 1799* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1890. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
yφ    &   (φ → Ⅎyψ)    &   (φ → (x = y → (ψχ)))       (φ → (xψyχ))

Theoremcbvaldva 1800* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))       (φ → (xψyχ))

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