Theorem List for Intuitionistic Logic Explorer - 1801-1900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | a16nf 1801* |
If there is only one element in the universe, then everything satisfies
Ⅎ. (Contributed by Mario Carneiro,
7-Oct-2016.)
|
⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
|
Theorem | 2albidv 1802* |
Formula-building rule for 2 existential quantifiers (deduction form).
(Contributed by NM, 4-Mar-1997.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 ↔ ∀𝑥∀𝑦𝜒)) |
|
Theorem | 2exbidv 1803* |
Formula-building rule for 2 existential quantifiers (deduction form).
(Contributed by NM, 1-May-1995.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒)) |
|
Theorem | 3exbidv 1804* |
Formula-building rule for 3 existential quantifiers (deduction form).
(Contributed by NM, 1-May-1995.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒)) |
|
Theorem | 4exbidv 1805* |
Formula-building rule for 4 existential quantifiers (deduction form).
(Contributed by NM, 3-Aug-1995.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)) |
|
Theorem | 19.9v 1806* |
Special case of Theorem 19.9 of [Margaris] p.
89. (Contributed by NM,
28-May-1995.) (Revised by NM, 21-May-2007.)
|
⊢ (∃𝑥𝜑 ↔ 𝜑) |
|
Theorem | exlimdd 1807 |
Existential elimination rule of natural deduction. (Contributed by
Mario Carneiro, 9-Feb-2017.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ Ⅎ𝑥𝜒
& ⊢ (𝜑 → ∃𝑥𝜓)
& ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | 19.21v 1808* |
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 (𝜑 → ∀𝑥𝜑) in 19.21 1527 via
the use of distinct variable conditions combined with ax-17 1471.
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 1960 derived from df-eu 1958. The "f" stands
for "not free in" which is less restrictive than "does
not occur in".
(Contributed by NM, 5-Aug-1993.)
|
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
|
Theorem | alrimiv 1809* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Contributed by NM,
5-Aug-1993.)
|
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜓) |
|
Theorem | alrimivv 1810* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Contributed by NM,
31-Jul-1995.)
|
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥∀𝑦𝜓) |
|
Theorem | alrimdv 1811* |
Deduction from Theorem 19.21 of [Margaris] p.
90. (Contributed by NM,
10-Feb-1997.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) |
|
Theorem | nfdv 1812* |
Apply the definition of not-free in a context. (Contributed by Mario
Carneiro, 11-Aug-2016.)
|
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) |
|
Theorem | 2ax17 1813* |
Quantification of two variables over a formula in which they do not
occur. (Contributed by Alan Sare, 12-Apr-2011.)
|
⊢ (𝜑 → ∀𝑥∀𝑦𝜑) |
|
Theorem | alimdv 1814* |
Deduction from Theorem 19.20 of [Margaris] p.
90. (Contributed by NM,
3-Apr-1994.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
|
Theorem | eximdv 1815* |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Contributed by NM,
27-Apr-1994.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) |
|
Theorem | 2alimdv 1816* |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Contributed by NM,
27-Apr-2004.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒)) |
|
Theorem | 2eximdv 1817* |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Contributed by NM,
3-Aug-1995.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒)) |
|
Theorem | 19.23v 1818* |
Special case of Theorem 19.23 of [Margaris] p.
90. (Contributed by NM,
28-Jun-1998.)
|
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) |
|
Theorem | 19.23vv 1819* |
Theorem 19.23 of [Margaris] p. 90 extended to
two variables.
(Contributed by NM, 10-Aug-2004.)
|
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) |
|
Theorem | sb56 1820* |
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 1700. (Contributed by
NM, 14-Apr-2008.)
|
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
|
Theorem | sb6 1821* |
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 = 𝑦 → 𝜑)) |
|
Theorem | sb5 1822* |
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 = 𝑦 ∧ 𝜑)) |
|
Theorem | sbnv 1823* |
Version of sbn 1881 where 𝑥 and 𝑦 are distinct.
(Contributed by
Jim Kingdon, 18-Dec-2017.)
|
⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
|
Theorem | sbanv 1824* |
Version of sban 1884 where 𝑥 and 𝑦 are distinct.
(Contributed by
Jim Kingdon, 24-Dec-2017.)
|
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) |
|
Theorem | sborv 1825* |
Version of sbor 1883 where 𝑥 and 𝑦 are distinct.
(Contributed by
Jim Kingdon, 3-Feb-2018.)
|
⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) |
|
Theorem | sbi1v 1826* |
Forward direction of sbimv 1828. (Contributed by Jim Kingdon,
25-Dec-2017.)
|
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
|
Theorem | sbi2v 1827* |
Reverse direction of sbimv 1828. (Contributed by Jim Kingdon,
18-Jan-2018.)
|
⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) |
|
Theorem | sbimv 1828* |
Intuitionistic proof of sbim 1882 where 𝑥 and 𝑦 are distinct.
(Contributed by Jim Kingdon, 18-Jan-2018.)
|
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
|
Theorem | sblimv 1829* |
Version of sblim 1886 where 𝑥 and 𝑦 are distinct.
(Contributed by
Jim Kingdon, 19-Jan-2018.)
|
⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)) |
|
Theorem | pm11.53 1830* |
Theorem *11.53 in [WhiteheadRussell]
p. 164. (Contributed by Andrew
Salmon, 24-May-2011.)
|
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
|
Theorem | exlimivv 1831* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Contributed by NM,
1-Aug-1995.)
|
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥∃𝑦𝜑 → 𝜓) |
|
Theorem | exlimdvv 1832* |
Deduction from Theorem 19.23 of [Margaris] p.
90. (Contributed by NM,
31-Jul-1995.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → 𝜒)) |
|
Theorem | exlimddv 1833* |
Existential elimination rule of natural deduction. (Contributed by
Mario Carneiro, 15-Jun-2016.)
|
⊢ (𝜑 → ∃𝑥𝜓)
& ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) |
|
Theorem | 19.27v 1834* |
Theorem 19.27 of [Margaris] p. 90.
(Contributed by NM, 3-Jun-2004.)
|
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) |
|
Theorem | 19.28v 1835* |
Theorem 19.28 of [Margaris] p. 90.
(Contributed by NM, 25-Mar-2004.)
|
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
|
Theorem | 19.36aiv 1836* |
Inference from Theorem 19.36 of [Margaris] p.
90. (Contributed by NM,
5-Aug-1993.)
|
⊢ ∃𝑥(𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) |
|
Theorem | 19.41v 1837* |
Special case of Theorem 19.41 of [Margaris] p.
90. (Contributed by NM,
5-Aug-1993.)
|
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
|
Theorem | 19.41vv 1838* |
Theorem 19.41 of [Margaris] p. 90 with 2
quantifiers. (Contributed by
NM, 30-Apr-1995.)
|
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓)) |
|
Theorem | 19.41vvv 1839* |
Theorem 19.41 of [Margaris] p. 90 with 3
quantifiers. (Contributed by
NM, 30-Apr-1995.)
|
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓)) |
|
Theorem | 19.41vvvv 1840* |
Theorem 19.41 of [Margaris] p. 90 with 4
quantifiers. (Contributed by
FL, 14-Jul-2007.)
|
⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑤∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓)) |
|
Theorem | 19.42v 1841* |
Special case of Theorem 19.42 of [Margaris] p.
90. (Contributed by NM,
5-Aug-1993.)
|
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) |
|
Theorem | exdistr 1842* |
Distribution of existential quantifiers. (Contributed by NM,
9-Mar-1995.)
|
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
|
Theorem | 19.42vv 1843* |
Theorem 19.42 of [Margaris] p. 90 with 2
quantifiers. (Contributed by
NM, 16-Mar-1995.)
|
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓)) |
|
Theorem | 19.42vvv 1844* |
Theorem 19.42 of [Margaris] p. 90 with 3
quantifiers. (Contributed by
NM, 21-Sep-2011.)
|
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓)) |
|
Theorem | 19.42vvvv 1845* |
Theorem 19.42 of [Margaris] p. 90 with 4
quantifiers. (Contributed by
Jim Kingdon, 23-Nov-2019.)
|
⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑤∃𝑥∃𝑦∃𝑧𝜓)) |
|
Theorem | exdistr2 1846* |
Distribution of existential quantifiers. (Contributed by NM,
17-Mar-1995.)
|
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓)) |
|
Theorem | 3exdistr 1847* |
Distribution of existential quantifiers. (Contributed by NM,
9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) |
|
Theorem | 4exdistr 1848* |
Distribution of existential quantifiers. (Contributed by NM,
9-Mar-1995.)
|
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃)))) |
|
Theorem | cbvalv 1849* |
Rule used to change bound variables, using implicit substitition.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
|
Theorem | cbvexv 1850* |
Rule used to change bound variables, using implicit substitition.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
|
Theorem | cbval2 1851* |
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.)
|
⊢ Ⅎ𝑧𝜑
& ⊢ Ⅎ𝑤𝜑
& ⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑦𝜓
& ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
|
Theorem | cbvex2 1852* |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro,
6-Oct-2016.)
|
⊢ Ⅎ𝑧𝜑
& ⊢ Ⅎ𝑤𝜑
& ⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑦𝜓
& ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
|
Theorem | cbval2v 1853* |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 4-Feb-2005.)
|
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
|
Theorem | cbvex2v 1854* |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 26-Jul-1995.)
|
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
|
Theorem | cbvald 1855* |
Deduction used to change bound variables, using implicit substitution,
particularly useful in conjunction with dvelim 1948. (Contributed by NM,
2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf
Lammen, 13-May-2018.)
|
⊢ Ⅎ𝑦𝜑
& ⊢ (𝜑 → Ⅎ𝑦𝜓)
& ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
|
Theorem | cbvexdh 1856* |
Deduction used to change bound variables, using implicit substitition,
particularly useful in conjunction with dvelim 1948. (Contributed by NM,
2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
|
⊢ (𝜑 → ∀𝑦𝜑)
& ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
|
Theorem | cbvexd 1857* |
Deduction used to change bound variables, using implicit substitution,
particularly useful in conjunction with dvelim 1948. (Contributed by NM,
2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten
by Jim Kingdon, 10-Jun-2018.)
|
⊢ Ⅎ𝑦𝜑
& ⊢ (𝜑 → Ⅎ𝑦𝜓)
& ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
|
Theorem | cbvaldva 1858* |
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) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
|
Theorem | cbvexdva 1859* |
Rule used to change the bound variable in an existential quantifier with
implicit substitution. Deduction form. (Contributed by David Moews,
1-May-2017.)
|
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
|
Theorem | cbvex4v 1860* |
Rule used to change bound variables, using implicit substitition.
(Contributed by NM, 26-Jul-1995.)
|
⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) & ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
|
Theorem | eean 1861 |
Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
(Revised by Mario Carneiro, 6-Oct-2016.)
|
⊢ Ⅎ𝑦𝜑
& ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
|
Theorem | eeanv 1862* |
Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
|
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
|
Theorem | eeeanv 1863* |
Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
(Proof shortened by Andrew Salmon, 25-May-2011.)
|
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒)) |
|
Theorem | ee4anv 1864* |
Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
|
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) |
|
Theorem | ee8anv 1865* |
Rearrange existential quantifiers. (Contributed by Jim Kingdon,
23-Nov-2019.)
|
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓)) |
|
Theorem | nexdv 1866* |
Deduction for generalization rule for negated wff. (Contributed by NM,
5-Aug-1993.)
|
⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → ¬ ∃𝑥𝜓) |
|
Theorem | chvarv 1867* |
Implicit substitution of 𝑦 for 𝑥 into a theorem.
(Contributed
by NM, 20-Apr-1994.)
|
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 |
|
Theorem | cleljust 1868* |
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 1446
with the
class variables in wcel 1445. (Contributed by NM, 28-Jan-2004.)
|
⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) |
|
1.4.5 More substitution theorems
|
|
Theorem | hbs1 1869* |
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) |
|
Theorem | nfs1v 1870* |
𝑥
is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are
distinct.
(Contributed by Mario Carneiro, 11-Aug-2016.)
|
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
|
Theorem | sbhb 1871* |
Two ways of expressing "𝑥 is (effectively) not free in 𝜑."
(Contributed by NM, 29-May-2009.)
|
⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)) |
|
Theorem | hbsbv 1872* |
This is a version of hbsb 1878 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 1873* |
Similar to hbsb 1878 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 1874* |
Closed form of nfsbxy 1873. (Contributed by Jim Kingdon, 9-May-2018.)
|
⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
|
Theorem | sbco2vlem 1875* |
This is a version of sbco2 1894 where 𝑧 is distinct from 𝑥 and
from
𝑦. It is a lemma on the way to proving
sbco2v 1876 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 | sbco2v 1876* |
This is a version of sbco2 1894 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 1877* |
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 | hbsb 1878* |
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 1879* |
Lemma for equsb3 1880. (Contributed by NM, 4-Dec-2005.) (Proof
shortened
by Andrew Salmon, 14-Jun-2011.)
|
⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) |
|
Theorem | equsb3 1880* |
Substitution applied to an atomic wff. (Contributed by Raph Levien and
FL, 4-Dec-2005.)
|
⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) |
|
Theorem | sbn 1881 |
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 1882 |
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 1883 |
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 1884 |
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 1885 |
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 1886 |
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 1887 |
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 1888 |
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 1889 |
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 1890 |
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 1891 |
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 1892* |
This is a version of sbco2 1894 where 𝑧 is distinct from 𝑦. It is
a lemma on the way to proving sbco2 1894 which has no distinct variable
constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
|
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
|
Theorem | sbco2h 1893 |
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 1894 |
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 1895 |
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 1896* |
Version of sbco2d 1895 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 1897 |
A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|
⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
|
Theorem | sbco3v 1898* |
Version of sbco3 1903 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 1899 |
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 1900* |
Version of sbcom 1904 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) |