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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hba1OLD 1801 | Obsolete proof of hba1 1800 as of 15-Dec-2017 (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nfa1 1802 |
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Theorem | a5i 1803 | Inference version of ax5o 1761. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfnf1 1804 |
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Theorem | nfnd 1805 |
If in a context ![]() ![]() ![]() ![]() |
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Theorem | nfndOLD 1806 | Obsolete proof of nfnd 1805 as of 28-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nfn 1807 |
If ![]() ![]() ![]() ![]() |
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Theorem | 19.38 1808 | Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 2-Jan-2018.) |
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Theorem | 19.21t 1809 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
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Theorem | 19.21 1810 |
Theorem 19.21 of [Margaris] p. 90. The
hypothesis can be thought of
as "![]() ![]() |
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Theorem | 19.21h 1811 |
Theorem 19.21 of [Margaris] p. 90. The
hypothesis can be thought of
as "![]() ![]() |
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Theorem | stdpc5 1812 |
An axiom scheme of standard predicate calculus that emulates Axiom 5 of
[Mendelson] p. 69. The hypothesis
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | stdpc5OLD 1813 | Obsolete proof of stdpc5 1812 as of 1-Jan-2018. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.23t 1814 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
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Theorem | 19.23 1815 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
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Theorem | 19.23h 1816 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
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Theorem | exlimi 1817 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | exlimih 1818 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
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Theorem | exlimihOLD 1819 | Obsolete proof of exlimih 1818 as of 1-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | exlimd 1820 | Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
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Theorem | exlimdOLD 1821 | Obsolete proof of exlimd 1820 as of 12-Jan-2018. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | exlimdh 1822 | Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.) |
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Theorem | nfimd 1823 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfimdOLD 1824 | Obsolete proof of nfimd 1823 as of 29-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | hbim1 1825 | A closed form of hbim 1832. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfim1 1826 | A closed form of nfim 1828. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
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Theorem | nfim1OLD 1827 | A closed form of nfim 1828. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nfim 1828 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfimOLD 1829 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbimd 1830 | Deduction form of bound-variable hypothesis builder hbim 1832. (Contributed by NM, 1-Jan-2002.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
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Theorem | hbimdOLD 1831 | Obsolete proof of hbimd 1830 as of 16-Dec-2017. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | hbim 1832 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbimOLD 1833 | Obsolete proof of hbim 1832 as of 1-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.23tOLD 1834 | Obsolete proof of 19.23t 1814 as of 1-Jan-2018. (Contributed by NM, 7-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.23hOLD 1835 | Obsolete proof of 19.23h 1816 as of 1-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | spimehOLD 1836* | Obsolete proof of spimeh 1675 as of 10-Dec-2017. (Contributed by NM, 7-Aug-1994.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.27 1837 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.28 1838 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfand 1839 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nf3and 1840 | Deduction form of bound-variable hypothesis builder nf3an 1845. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) |
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Theorem | nfan1 1841 | A closed form of nfan 1842. (Contributed by Mario Carneiro, 3-Oct-2016.) |
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Theorem | nfan 1842 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfnan 1843 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfanOLD 1844 | Obsolete proof of nfan 1842 as of 2-Jan-2018. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nf3an 1845 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hban 1846 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbanOLD 1847 | Obsolete proof of hban 1846 as of 2-Jan-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | hb3an 1848 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hb3anOLD 1849 | Obsolete proof of hb3an 1848 as of 2-Jan-2018. (Contributed by NM, 14-Sep-2003.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nfbid 1850 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfbidOLD 1851 | Obsolete proof of nfbid 1850 as of 29-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nfbi 1852 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfbiOLD 1853 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfor 1854 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nf3or 1855 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | equsalhw 1856* | Weaker version of equsalh 1968 (requiring distinct variables) without using ax-12 1946. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) |
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Theorem | equsalhwOLD 1857* | Obsolete proof of equsalhw 1856 as of 28-Dec-2017. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.21hOLD 1858 | Obsolete proof of 19.21h 1811 as of 1-Jan-2018. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | hbex 1859 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfal 1860 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfex 1861 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfexOLD 1862 | Obsolete proof of nfex 1861 as of 30-Dec-2017. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nfnf 1863 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfnfOLD 1864 | Obsolete proof of nfnf 1863 as of 30-Dec-2017. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.12 1865 | Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 1917 and r19.12sn 3832. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
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Theorem | 19.12OLD 1866 | Obsolete proof of 19.12 1865 as of 3-Jan-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nfald 1867 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfaldOLD 1868 | Obsolete proof of nfald 1867 as of 6-Jan-2018. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | nfexd 1869 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfa2 1870 | Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | nfia1 1871 | Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | dvelimhw 1872* | Proof of dvelimh 2015 without using ax-12 1946 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) |
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Theorem | dvelimhwOLD 1873* | Proof of dvelimh 2015 without using ax-12 1946 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cbv3hv 1874* | Lemma for ax10 1991. Similar to cbv3h 2036. Requires distinct variables but avoids ax-12 1946. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) |
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Theorem | cbv3hvOLD 1875* | Obsolete proof of cbv3hv 1874 as of 29-Dec-2017. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.9tOLD 1876 | Obsolete proof of 19.9t 1789 as of 30-Dec-2017. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | excomimOLD 1877 | Obsolete proof of excomim 1753 as of 8-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | excomOLD 1878 | Obsolete proof of excom 1752 as of 8-Jan-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.16 1879 | Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.17 1880 | Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.19 1881 | Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.21tOLD 1882 | Obsolete proof of 19.21t 1809 as of 30-Dec-2017. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.21-2 1883 | Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.) |
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Theorem | 19.21bbi 1884 | Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.) |
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Theorem | nf2 1885 | An alternative definition of df-nf 1551, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | nf3 1886 | An alternative definition of df-nf 1551. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | nf4 1887 |
Variable ![]() ![]() ![]() |
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Theorem | 19.36 1888 | Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.36i 1889 | Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.37 1890 | Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.38OLD 1891 | Obsolete proof of 19.38 1808 as of 2-Jan-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.32 1892 | Theorem 19.32 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
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Theorem | 19.31 1893 | Theorem 19.31 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.44 1894 | Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.45 1895 | Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.41 1896 | Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
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Theorem | 19.41OLD 1897 | Obsolete proof of 19.41 1896 as of 12-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 19.42 1898 | Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
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Theorem | exan 1899 | Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) |
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Theorem | exanOLD 1900 | Obsolete proof of exan 1899 as of 13-Jan-2018. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
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