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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sb4a 1801 | A version of sb4 1832 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
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Theorem | equs45f 1802 |
Two ways of expressing substitution when ![]() ![]() |
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Theorem | sb6f 1803 |
Equivalence for substitution when ![]() ![]() |
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Theorem | sb5f 1804 |
Equivalence for substitution when ![]() ![]() |
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Theorem | sb4e 1805 | One direction of a simplified definition of substitution that unlike sb4 1832 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
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Theorem | hbsb2a 1806 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
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Theorem | hbsb2e 1807 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
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Theorem | hbsb3 1808 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfs1 1809 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sbcof2 1810 |
Version of sbco 1968 where ![]() ![]() |
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Theorem | spimv 1811* | A version of spim 1738 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.) |
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Theorem | aev 1812* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1814. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
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Theorem | ax16 1813* |
Theorem showing that ax-16 1814 is redundant if ax-17 1526 is included in the
axiom system. The important part of the proof is provided by aev 1812.
See ax16ALT 1859 for an alternate proof that does not require ax-10 1505 or ax12 1512. This theorem should not be referenced in any proof. Instead, use ax-16 1814 below so that theorems needing ax-16 1814 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
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Axiom | ax-16 1814* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1526 to be a
metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p.
16 of the preprint). It apparently does not otherwise appear in the
literature but is easily proved from textbook predicate calculus by
cases. It is a somewhat bizarre axiom since the antecedent is always
false in set theory, but nonetheless it is technically necessary as you
can see from its uses.
This axiom is redundant if we include ax-17 1526; see Theorem ax16 1813. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1813. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
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Theorem | dveeq2 1815* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
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Theorem | dveeq2or 1816* |
Quantifier introduction when one pair of variables is distinct. Like
dveeq2 1815 but connecting ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvelimfALT2 1817* | Proof of dvelimf 2015 using dveeq2 1815 (shown as the last hypothesis) instead of ax12 1512. This shows that ax12 1512 could be replaced by dveeq2 1815 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
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Theorem | nd5 1818* | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.) |
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Theorem | exlimdv 1819* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
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Theorem | ax11v2 1820* |
Recovery of ax11o 1822 from ax11v 1827 without using ax-11 1506. The hypothesis
is even weaker than ax11v 1827, with ![]() ![]() ![]() |
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Theorem | ax11a2 1821* |
Derive ax-11o 1823 from a hypothesis in the form of ax-11 1506. The
hypothesis is even weaker than ax-11 1506, with ![]() ![]() ![]() |
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Theorem | ax11o 1822 |
Derivation of set.mm's original ax-11o 1823 from the shorter ax-11 1506 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1814 or ax-17 1526. Normally, ax11o 1822 should be used rather than ax-11o 1823, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.) |
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Axiom | ax-11o 1823 |
Axiom ax-11o 1823 ("o" for "old") was the
original version of ax-11 1506,
before it was discovered (in Jan. 2007) that the shorter ax-11 1506 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 "![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() This axiom is redundant, as shown by Theorem ax11o 1822. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1822. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
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Theorem | albidv 1824* | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
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Theorem | exbidv 1825* | Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
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Theorem | ax11b 1826 | A bidirectional version of ax-11o 1823. (Contributed by NM, 30-Jun-2006.) |
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Theorem | ax11v 1827* | This is a version of ax-11o 1823 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.) |
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Theorem | ax11ev 1828* | Analogue to ax11v 1827 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.) |
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Theorem | equs5 1829 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) |
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Theorem | equs5or 1830 | Lemma used in proofs of substitution properties. Like equs5 1829 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.) |
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Theorem | sb3 1831 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sb4 1832 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sb4or 1833 | One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1832 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.) |
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Theorem | sb4b 1834 | Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
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Theorem | sb4bor 1835 | Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.) |
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Theorem | hbsb2 1836 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfsb2or 1837 | Bound-variable hypothesis builder for substitution. Similar to hbsb2 1836 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.) |
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Theorem | sbequilem 1838 | Propositional logic lemma used in the sbequi 1839 proof. (Contributed by Jim Kingdon, 1-Feb-2018.) |
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Theorem | sbequi 1839 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.) |
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Theorem | sbequ 1840 | 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.) |
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Theorem | drsb2 1841 | 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.) |
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Theorem | spsbe 1842 | 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.) |
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Theorem | spsbim 1843 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
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Theorem | spsbbi 1844 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
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Theorem | sbbidh 1845 | Deduction substituting both sides of a biconditional. New proofs should use sbbid 1846 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
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Theorem | sbbid 1846 | Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) |
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Theorem | sbequ8 1847 | Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.) |
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Theorem | sbft 1848 | Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) |
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Theorem | sbid2h 1849 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sbid2 1850 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
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Theorem | sbidm 1851 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
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Theorem | sb5rf 1852 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | sb6rf 1853 | Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | sb8h 1854 | 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.) |
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Theorem | sb8eh 1855 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.) |
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Theorem | sb8 1856 | 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.) |
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Theorem | sb8e 1857 | 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.) |
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Theorem | ax16i 1858* | Inference with ax-16 1814 as its conclusion, that does not require ax-10 1505, ax-11 1506, or ax12 1512 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.) |
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Theorem | ax16ALT 1859* | Version of ax16 1813 that does not require ax-10 1505 or ax12 1512 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | spv 1860* | Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.) |
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Theorem | spimev 1861* | Distinct-variable version of spime 1741. (Contributed by NM, 5-Aug-1993.) |
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Theorem | speiv 1862* | Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.) |
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Theorem | equvin 1863* | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.) |
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Theorem | a16g 1864* | A generalization of Axiom ax-16 1814. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | a16gb 1865* | A generalization of Axiom ax-16 1814. (Contributed by NM, 5-Aug-1993.) |
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Theorem | a16nf 1866* |
If there is only one element in the universe, then everything satisfies
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Theorem | 2albidv 1867* | Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.) |
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Theorem | 2exbidv 1868* | Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.) |
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Theorem | 3exbidv 1869* | Formula-building rule for 3 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.) |
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Theorem | 4exbidv 1870* | Formula-building rule for 4 existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.) |
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Theorem | 19.9v 1871* | Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.) |
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Theorem | exlimdd 1872 | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
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Theorem | 19.21v 1873* |
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | alrimiv 1874* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alrimivv 1875* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
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Theorem | alrimdv 1876* | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) |
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Theorem | nfdv 1877* | Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | 2ax17 1878* | Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.) |
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Theorem | alimdv 1879* | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.) |
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Theorem | eximdv 1880* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
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Theorem | 2alimdv 1881* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.) |
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Theorem | 2eximdv 1882* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.) |
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Theorem | 19.23v 1883* | Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.) |
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Theorem | 19.23vv 1884* | Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.) |
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Theorem | sb56 1885* |
Two equivalent ways of expressing the proper substitution of ![]() ![]() ![]() ![]() ![]() |
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Theorem | sb6 1886* | 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.) |
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Theorem | sb5 1887* | Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.) |
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Theorem | sbnv 1888* |
Version of sbn 1952 where ![]() ![]() |
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Theorem | sbanv 1889* |
Version of sban 1955 where ![]() ![]() |
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Theorem | sborv 1890* |
Version of sbor 1954 where ![]() ![]() |
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Theorem | sbi1v 1891* | Forward direction of sbimv 1893. (Contributed by Jim Kingdon, 25-Dec-2017.) |
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Theorem | sbi2v 1892* | Reverse direction of sbimv 1893. (Contributed by Jim Kingdon, 18-Jan-2018.) |
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Theorem | sbimv 1893* |
Intuitionistic proof of sbim 1953 where ![]() ![]() |
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Theorem | sblimv 1894* |
Version of sblim 1957 where ![]() ![]() |
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Theorem | pm11.53 1895* | Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | exlimivv 1896* | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.) |
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Theorem | exlimdvv 1897* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
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Theorem | exlimddv 1898* | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.) |
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Theorem | 19.27v 1899* | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.) |
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Theorem | 19.28v 1900* | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.) |
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