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Theorem List for Intuitionistic Logic Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremalrimdv 1801* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremnfdv 1802* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theorem2ax17 1803* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(𝜑 → ∀𝑥𝑦𝜑)

Theoremalimdv 1804* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremeximdv 1805* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theorem2alimdv 1806* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))

Theorem2eximdv 1807* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 → ∃𝑥𝑦𝜒))

Theorem19.23v 1808* Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
(∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.23vv 1809* Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theoremsb56 1810* 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 1690. (Contributed by NM, 14-Apr-2008.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb6 1811* 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]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb5 1812* 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]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremsbnv 1813* Version of sbn 1871 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Theoremsbanv 1814* Version of sban 1874 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Theoremsborv 1815* Version of sbor 1873 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))

Theoremsbi1v 1816* Forward direction of sbimv 1818. (Contributed by Jim Kingdon, 25-Dec-2017.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbi2v 1817* Reverse direction of sbimv 1818. (Contributed by Jim Kingdon, 18-Jan-2018.)
(([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))

Theoremsbimv 1818* Intuitionistic proof of sbim 1872 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsblimv 1819* Version of sblim 1876 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
(𝜓 → ∀𝑥𝜓)       ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))

Theorempm11.53 1820* Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))

Theoremexlimivv 1821* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑𝜓)

Theoremexlimdvv 1822* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓𝜒))

Theoremexlimddv 1823* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theorem19.27v 1824* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28v 1825* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.36aiv 1826* Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem19.41v 1827* Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.41vv 1828* Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theorem19.41vvv 1829* Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))

Theorem19.41vvvv 1830* Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))

Theorem19.42v 1831* Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theoremexdistr 1832* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theorem19.42vv 1833* Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝜓))

Theorem19.42vvv 1834* Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))

Theorem19.42vvvv 1835* Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))

Theoremexdistr2 1836* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))

Theorem3exdistr 1837* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))

Theorem4exdistr 1838* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))

Theoremcbvalv 1839* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexv 1840* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbval2 1841* 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 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2 1842* 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 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbval2v 1843* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2v 1844* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvald 1845* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1938. (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 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexdh 1846* Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1938. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbvexd 1847* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1938. (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 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbvaldva 1848* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexdva 1849* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbvex4v 1850* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)

Theoremeean 1851 Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Theoremeeanv 1852* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Theoremeeeanv 1853* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))

Theoremee4anv 1854* Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
(∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))

Theoremee8anv 1855* Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
(∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝑤𝜑 ∧ ∃𝑣𝑢𝑡𝑠𝜓))

Theoremnexdv 1856* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremchvarv 1857* Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by NM, 20-Apr-1994.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremcleljust 1858* When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1437 with the class variables in wcel 1436. (Contributed by NM, 28-Jan-2004.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))

1.4.5  More substitution theorems

Theoremhbs1 1859* 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremnfs1v 1860* 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥[𝑦 / 𝑥]𝜑

Theoremsbhb 1861* Two ways of expressing "𝑥 is (effectively) not free in 𝜑." (Contributed by NM, 29-May-2009.)
((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))

Theoremhbsbv 1862* This is a version of hbsb 1868 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)

Theoremnfsbxy 1863* Similar to hbsb 1868 but with an extra distinct variable constraint, on 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Mar-2018.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑

Theoremnfsbxyt 1864* Closed form of nfsbxy 1863. (Contributed by Jim Kingdon, 9-May-2018.)
(∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Theoremsbco2vlem 1865* This is a version of sbco2 1884 where 𝑧 is distinct from 𝑥 and from 𝑦. It is a lemma on the way to proving sbco2v 1866 which only requires that 𝑧 and 𝑥 be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) (One distinct variable constraint removed by Jim Kingdon, 3-Feb-2018.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbco2v 1866* This is a version of sbco2 1884 where 𝑧 is distinct from 𝑥. (Contributed by Jim Kingdon, 12-Feb-2018.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremnfsb 1867* 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.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑

Theoremhbsb 1868* 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)

Theoremequsb3lem 1869* Lemma for equsb3 1870. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)

Theoremequsb3 1870* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)

Theoremsbn 1871 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)

Theoremsbim 1872 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbor 1873 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](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))

Theoremsban 1874 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](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Theoremsbrim 1875 Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsblim 1876 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.)
𝑥𝜓       ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))

Theoremsb3an 1877 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
([𝑦 / 𝑥](𝜑𝜓𝜒) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓 ∧ [𝑦 / 𝑥]𝜒))

Theoremsbbi 1878 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

Theoremsblbis 1879 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜒𝜑) ↔ ([𝑦 / 𝑥]𝜒𝜓))

Theoremsbrbis 1880 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜑𝜒) ↔ (𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbrbif 1881 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
(𝜒 → ∀𝑥𝜒)    &   ([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜑𝜒) ↔ (𝜓𝜒))

Theoremsbco2yz 1882* This is a version of sbco2 1884 where 𝑧 is distinct from 𝑦. It is a lemma on the way to proving sbco2 1884 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
𝑧𝜑       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbco2h 1883 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)

Theoremsbco2 1884 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑧𝜑       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbco2d 1885 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))

Theoremsbco2vd 1886* Version of sbco2d 1885 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))

Theoremsbco 1887 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbco3v 1888* Version of sbco3 1893 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)

Theoremsbcocom 1889 Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)

Theoremsbcomv 1890* Version of sbcom 1894 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)

Theoremsbcomxyyz 1891* Version of sbcom 1894 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)

Theoremsbco3xzyz 1892* Version of sbco3 1893 with distinct variable constraints between 𝑥 and 𝑧, and 𝑦 and 𝑧. Lemma for proving sbco3 1893. (Contributed by Jim Kingdon, 22-Mar-2018.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Theoremsbco3 1893 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)

Theoremsbcom 1894 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)

Theoremnfsbt 1895* Closed form of nfsb 1867. (Contributed by Jim Kingdon, 9-May-2018.)
(∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Theoremnfsbd 1896* Deduction version of nfsb 1867. (Contributed by NM, 15-Feb-2013.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Theoremelsb3 1897* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)

Theoremelsb4 1898* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)

Theoremsb9v 1899* Like sb9 1900 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)

Theoremsb9 1900 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)

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