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

Theorema4sd 1701 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)

Theoremnfr 1702 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)

Theoremnfri 1703 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfrd 1704 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremalimd 1705 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremalrimi 1706 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfd 1707 Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfdh 1708 Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremalrimdd 1709 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremalrimd 1710 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremeximd 1711 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnexd 1712 Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremalbid 1713 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremexbid 1714 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfbidf 1715 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theorema6e 1716 Abbreviated version of ax-6o 1697. (Contributed by NM, 5-Aug-1993.)

Theoremhbnt 1717 Closed theorem version of bound-variable hypothesis builder hbn 1722. (Contributed by NM, 5-Aug-1993.)

Theoremhba1 1718 is not free in . Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.)

Theoremnfa1 1719 is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfnf1 1720 is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theorema5i 1721 Inference version of ax-5o 1694. (Contributed by NM, 5-Aug-1993.)

Theoremhbn 1722 If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremhbim 1723 If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.)

Theoremhban 1724 If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremhb3an 1725 If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.)

Theoremnfnd 1726 If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfimd 1727 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfbid 1728 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfand 1729 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremnf3and 1730 Deduction form of bound-variable hypothesis builder nf3an 1740. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)

Theoremnfn 1731 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfal 1732 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfex 1733 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfnf 1734 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfim 1735 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfor 1736 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfan 1737 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfbi 1738 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnf3or 1739 If is not free in , , and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnf3an 1740 If is not free in , , and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfnth 1741 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)

Theoremnfald 1742 If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfexd 1743 If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfa2 1744 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfia1 1745 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremax46 1746 Proof of a single axiom that can replace ax-4 1692 and ax-6o 1697. See ax46to4 1747 and ax46to6 1748 for the re-derivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.)

Theoremax46to4 1747 Re-derivation of ax-4 1692 from ax46 1746. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.)

Theoremax46to6 1748 Re-derivation of ax-6o 1697 from ax46 1746. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.)

Theoremax67 1749 Proof of a single axiom that can replace both ax-6o 1697 and ax-7 1535. See ax67to6 1750 and ax67to7 1751 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.)

Theoremax67to6 1750 Re-derivation of ax-6o 1697 from ax67 1749. Note that ax-6o 1697 and ax-7 1535 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.)

Theoremax67to7 1751 Re-derivation of ax-7 1535 from ax67 1749. Note that ax-6o 1697 and ax-7 1535 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.)

Theoremax467 1752 Proof of a single axiom that can replace ax-4 1692, ax-6o 1697, and ax-7 1535 in a subsystem that includes these axioms plus ax-5o 1694 and ax-gen 1536 (and propositional calculus). See ax467to4 1753, ax467to6 1754, and ax467to7 1755 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 1746. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.)

Theoremax467to4 1753 Re-derivation of ax-4 1692 from ax467 1752. Only propositional calculus is used by the re-derivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.)

Theoremax467to6 1754 Re-derivation of ax-6o 1697 from ax467 1752. Note that ax-6o 1697 and ax-7 1535 are not used by the re-derivation. The use of alimi 1546 (which uses ax-4 1692) is allowed since we have already proved ax467to4 1753. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.)

Theoremax467to7 1755 Re-derivation of ax-7 1535 from ax467 1752. Note that ax-6o 1697 and ax-7 1535 are not used by the re-derivation. The use of alimi 1546 (which uses ax-4 1692) is allowed since we have already proved ax467to4 1753. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.)

Theoremmodal-5 1756 The analog in our "pure" predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)

Theoremmodal-b 1757 The analog in our "pure" predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)

Theorem19.8a 1758 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)

Theorem19.2 1759 Theorem 19.2 of [Margaris] p. 89, generalized to use two set variables. (Contributed by O'Cat, 31-Mar-2008.)

Theorem19.3 1760 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.9t 1761 A closed version of 19.9 1762. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.9 1762 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.9d 1763 A deduction version of one direction of 19.9 1762. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theoremexcomim 1764 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theoremexcom 1765 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)

Theorem19.12 1766 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 2031 and r19.12sn 3600. (Contributed by NM, 5-Aug-1993.)

Theorem19.16 1767 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.17 1768 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.19 1769 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.21t 1770 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.21 1771 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " is not free in ." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.21-2 1772 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)

Theoremstdpc5 1773 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis can be thought of as emulating " is not free in ." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example would not (for us) be free in by nfequid 1688. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theorem19.21bi 1774 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.21bbi 1775 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)

Theorem19.23t 1776 Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.)

Theorem19.23 1777 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theoremnf2 1778 An alternative definition of df-nf 1540, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnf3 1779 An alternative definition of df-nf 1540. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnf4 1780 Variable is effectively not free in iff is always true or always false. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremexlimi 1781 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremexlimih 1782 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theorem19.23bi 1783 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremexlimd 1784 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremexlimdh 1785 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)

Theorem19.27 1786 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.28 1787 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.36 1788 Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.36i 1789 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.37 1790 Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.38 1791 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.39 1792 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.24 1793 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.32 1794 Theorem 19.32 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.31 1795 Theorem 19.31 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.44 1796 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.45 1797 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.34 1798 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.41 1799 Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem19.42 1800 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

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