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Type | Label | Description |
---|---|---|

Statement | ||

Theorem | ax12b 1701 | Two equivalent ways of expressing ax-12 1950. See the comment for ax-12 1950. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) |

Theorem | ax12bOLD 1702 | Obsolete version of ax12b 1701 as of 12-Aug-2017. (Contributed by NM, 2-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | spfw 1703* | Weak version of sp 1763. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. TO DO: Do we need this theorem? If not, maybe it should be deleted. (Contributed by NM, 19-Apr-2017.) |

Theorem | spnfwOLD 1704 | Weak version of sp 1763. Uses only Tarski's FOL axiom schemes. Obsolete version of spnfw 1682 as of 13-Aug-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.8wOLD 1705 | Obsolete version of 19.8w 1672 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | spw 1706* | Weak version of specialization scheme sp 1763. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1763 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1763 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1738 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1763 are spfw 1703 (minimal distinct variable requirements), spnfw 1682 (when is not free in ), spvw 1678 (when does not appear in ), sptruw 1683 (when is true), and spfalw 1684 (when is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.) |

Theorem | spwOLD 1707* | Obsolete proof of spw 1706 as of 27-Feb-2018. (Contributed by NM, 9-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | spvwOLD 1708* | Obsolete version of spvw 1678 as of 4-Dec-2017. (Contributed by NM, 10-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.3vOLD 1709* | Obsolete version of 19.3v 1677 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.9vOLD 1710* | Obsolete version of 19.9v 1676 as of 4-Dec-2017. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | exlimivOLD 1711* | Obsolete version of exlimiv 1644 as of 4-Dec-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | spfalwOLD 1712 | Obsolete proof of spfalw 1684 as of 25-Dec-2017. (Contributed by NM, 23-Apr-1017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.2OLD 1713 | Obsolete version of 19.2 1671 as of 4-Dec-2017. (Contributed by NM, 2-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | cbvalw 1714* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | cbvalvw 1715* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.) |

Theorem | cbvalvwOLD 1716* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | cbvexvw 1717* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |

Theorem | alcomiw 1718* | Weak version of alcom 1752. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) |

Theorem | hbn1fw 1719* | Weak version of ax-6 1744 from which we can prove any ax-6 1744 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.) |

Theorem | hbn1fwOLD 1720* | Obsolete proof of hbn1fw 1719 as of 28-Feb-2018. (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | hbn1w 1721* | Weak version of hbn1 1745. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | hba1w 1722* | Weak version of hba1 1804. See comments for ax6w 1732. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | hbe1w 1723* | Weak version of hbe1 1746. See comments for ax6w 1732. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |

Theorem | hbalw 1724* | Weak version of hbal 1751. Uses only Tarski's FOL axiom schemes. Unlike hbal 1751, this theorem requires that and be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.) |

1.4.8 Membership predicate | ||

Syntax | wcel 1725 |
Extend wff definition to include the membership connective between
classes.
For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing here is to allow us to express i.e. "prove" the wel 1726 of predicate calculus in terms of the wceq 1652 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2422 for more information on the set theory usage of wcel 1725.) |

Theorem | wel 1726 |
Extend wff definition to include atomic formulas with the epsilon
(membership) predicate. This is read " is an element of
," " is a member of ," " belongs to ,"
or " contains
." Note: The
phrase " includes
" means
" is a subset of
;" to use it also
for
, as some authors occasionally do, is poor form and causes
confusion, according to George Boolos (1992 lecture at MIT).
This syntactical construction introduces a binary non-logical predicate symbol (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. (Instead of introducing wel 1726 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1725. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1726 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1725. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |

1.4.9 Axiom scheme ax-13 (Left Equality for
Binary Predicate) | ||

Axiom | ax-13 1727 | Axiom of Left Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of an arbitrary binary predicate , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.) |

Theorem | elequ1 1728 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |

1.4.10 Axiom scheme ax-14 (Right Equality for
Binary Predicate) | ||

Axiom | ax-14 1729 | Axiom of Right Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of an arbitrary binary predicate , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |

Theorem | elequ2 1730 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |

1.4.11 Logical redundancy of ax-6 , ax-7 , ax-11
, ax-12The orginal axiom schemes of Tarski's predicate calculus are ax-5 1566, ax-17 1626, ax9v 1667, ax-8 1687, ax-13 1727, and ax-14 1729, together with rule ax-gen 1555. See http://us.metamath.org/mpeuni/mmset.html#compare 1555. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85. The axiom system of set.mm includes the auxiliary axiom schemes ax-6 1744, ax-7 1749, ax-12 1950, and ax-11 1761, which are not part of Tarski's axiom schemes. They are used (and we conjecture are required) to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and set metavariables, bundled or not, whose object-language instances are valid. (ax-11 1761 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.) (There are additional predicate calculus axiom schemes included in set.mm such as ax-4 2211, but they can all be proved as theorems from the above.) Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the and in ax-9 1666 are bundled, but they are not in ax9v 1667. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax9v 1667 is the principal instance of ax-9 1666. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance of ax-9 1666 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with. There is also a small economy in being able to state principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them). Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-6 1744, ax-7 1749, ax-11 1761, and ax-12 1950 . "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-6 1744, ax-7 1749, ax-11 1761, or ax-12 1950 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.) It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes. The new theorem schemes ax6w 1732, ax7w 1733, ax11w 1736, and ax12w 1739 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-6 1744, ax-7 1749, ax-11 1761, and ax-12 1950 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax6w 1732, ax7w 1733, and ax11w 1736 is of the form where is an auxiliary or "dummy" wff metavariable in which doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting conditions (1) and (2). The example ax11wdemo 1738 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this.
We also show the degenerate instances for axioms with bundled variables in
ax7dgen 1734, ax11dgen 1737, ax12dgen1 1740, ax12dgen2 1741, ax12dgen3 1742, and
ax12dgen4 1743. (Their proofs are trivial, but we include
them to be thorough.)
Combining the principal and degenerate cases It is interesting that Tarski used the bundled scheme ax-9 1666 in an older system, so it seems the main purpose of his later ax9v 1667 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-9 1666 as our official axiom, we show that the degenerate instance holds in ax9dgen 1731. The case of sp 1763 is curious: originally an axiom of Tarski's system, it was proved redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the compact scheme form apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 1706, again requiring substitution instances of that meet conditions (1) and (2) above. Note that our direct proof sp 1763 requires ax-11 1761, which is not part of Tarski's system. | ||

Theorem | ax9dgen 1731 | Tarski's system uses the weaker ax9v 1667 instead of the bundled ax-9 1666, so here we show that the degenerate case of ax-9 1666 can be derived. (Contributed by NM, 23-Apr-2017.) |

Theorem | ax6w 1732* | Weak version of ax-6 1744 from which we can prove any ax-6 1744 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | ax7w 1733* | Weak version of ax-7 1749 from which we can prove any ax-7 1749 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1749, this theorem requires that and be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.) |

Theorem | ax7dgen 1734 | Degenerate instance of ax-7 1749 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax11wlem 1735* | Lemma for weak version of ax-11 1761. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax11w 1736. (Contributed by NM, 10-Apr-2017.) |

Theorem | ax11w 1736* | Weak version of ax-11 1761 from which we can prove any ax-11 1761 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that and be distinct (unless does not occur in ). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for , see ax11wdemo 1738. (Contributed by NM, 10-Apr-2017.) |

Theorem | ax11dgen 1737 | Degenerate instance of ax-11 1761 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax11wdemo 1738* | Example of an application of ax11w 1736 that results in an instance of ax-11 1761 for a contrived formula with mixed free and bound variables, , in place of . The proof illustrates bound variable renaming with cbvalvw 1715 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.) |

Theorem | ax12w 1739* | Weak version (principal instance) of ax-12 1950. (Because and don't need to be distinct, this actually bundles the principal instance and the degenerate instance .) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax6w 1732, ax7w 1733, and ax11w 1736. (Contributed by NM, 10-Apr-2017.) |

Theorem | ax12dgen1 1740 | Degenerate instance of ax-12 1950 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax12dgen2 1741 | Degenerate instance of ax-12 1950 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax12dgen3 1742 | Degenerate instance of ax-12 1950 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax12dgen4 1743 | Degenerate instance of ax-12 1950 where bundled variables , , and have a common substitution. Uses only Tarski's FOL axiom schemes . (Contributed by NM, 13-Apr-2017.) |

1.5 Predicate calculus with equality: Auxiliary
axiom schemes (4 schemes)In this section we introduce four additional schemes ax-6 1744, ax-7 1749, ax-11 1761, and ax-12 1950 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "metalogical completeness," which means that we can prove (with Metamath) all possible schemes expressible in our language of wff metavariables ranging over object-language wffs and set metavariables ranging over object-language individual variables. To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax6w 1732, ax7w 1733, ax12w 1739, and ax11w 1736, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all set metavariables to be mutually distinct) can be proved using only the schemes in Tarski's system S2.
An open problem is to show that these four additional schemes are mutually
| ||

1.5.1 Axiom scheme ax-6 (Quantified
Negation) | ||

Axiom | ax-6 1744 | Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax6w 1732) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.) |

Theorem | hbn1 1745 | is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.) |

Theorem | hbe1 1746 | is not free in . (Contributed by NM, 5-Aug-1993.) |

Theorem | nfe1 1747 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | modal-5 1748 | The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.) |

1.5.2 Axiom scheme ax-7 (Quantifier
Commutation) | ||

Axiom | ax-7 1749 | Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax7w 1733) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.) |

Theorem | a7s 1750 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |

Theorem | hbal 1751 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) |

Theorem | alcom 1752 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |

Theorem | alrot3 1753 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |

Theorem | alrot4 1754 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |

Theorem | hbald 1755 | Deduction form of bound-variable hypothesis builder hbal 1751. (Contributed by NM, 2-Jan-2002.) |

Theorem | excom 1756 | Theorem 19.11 of [Margaris] p. 89. Revised to remove dependency on ax-11 1761, ax-6 1744, ax-9 1666, ax-8 1687 and ax-17 1626. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 8-Jan-2018.) |

Theorem | excomim 1757 | One direction of Theorem 19.11 of [Margaris] p. 89. Revised to remove dependency on ax-11 1761, ax-6 1744, ax-9 1666, ax-8 1687 and ax-17 1626. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Revised by Wolf Lammen, 8-Jan-2018.) |

Theorem | excom13 1758 | Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |

Theorem | exrot3 1759 | Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) |

Theorem | exrot4 1760 | Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) |

1.5.3 Axiom scheme ax-11
(Substitution) | ||

Axiom | ax-11 1761 |
Axiom of Substitution. One of the 5 equality axioms of predicate
calculus. The final consequent
is a way of
expressing "
substituted for in wff
" (cf. sb6 2174).
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.
The original version of this axiom was ax-11o 2217 ("o" for "old") and was replaced with this shorter ax-11 1761 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 2077. Conversely, this axiom is proved from ax-11o 2217 as theorem ax11 2231. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2217) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. See ax11v 2171 and ax11v2 2074 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax11w 1736) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.) |

Theorem | 19.8a 1762 | 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.) (Revised by Wolf Lammen, 13-Jan-2018.) |

Theorem | sp 1763 |
Specialization. A universally quantified wff implies the wff without a
quantifier Axiom scheme B5 of [Tarski]
p. 67 (under his system S2,
defined in the last paragraph on p. 77). Also appears as Axiom scheme
C5' in [Megill] p. 448 (p. 16 of the
preprint).
For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2087. This theorem shows that our obsolete axiom ax-4 2211 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114. It appears that this scheme cannot be derived directly from Tarski's axioms without auxilliary axiom scheme ax-11 1761. It is thought the best we can do using only Tarski's axioms is spw 1706. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) |

Theorem | spOLD 1764 | Obsolete proof of sp 1763 as of 23-Dec-2017. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | ax5o 1765 |
Show that the original axiom ax-5o 2212 can be derived from ax-5 1566
and
others. See ax5 2222 for the rederivation of ax-5 1566
from ax-5o 2212.
Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.) |

Theorem | ax6o 1766 |
Show that the original axiom ax-6o 2213 can be derived from ax-6 1744
and
others. See ax6 2223 for the rederivation of ax-6 1744
from ax-6o 2213.
Normally, ax6o 1766 should be used rather than ax-6o 2213, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.) |

Theorem | a6e 1767 | Abbreviated version of ax6o 1766. (Contributed by NM, 5-Aug-1993.) |

Theorem | modal-b 1768 | The analog in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.) |

Theorem | spi 1769 | Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.) |

Theorem | sps 1770 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |

Theorem | spsd 1771 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |

Theorem | 19.8aOLD 1772 | 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.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.2g 1773 | Theorem 19.2 of [Margaris] p. 89, generalized to use two set variables. (Contributed by O'Cat, 31-Mar-2008.) |

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

Theorem | 19.23bi 1775 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | nexr 1776 | Inference from 19.8a 1762. (Contributed by Jeff Hankins, 26-Jul-2009.) |

Theorem | nfr 1777 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |

Theorem | nfri 1778 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfrd 1779 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | alimd 1780 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | alrimi 1781 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfd 1782 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfdh 1783 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | alrimdd 1784 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | alrimd 1785 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | eximd 1786 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nexd 1787 | Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | albid 1788 | Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | exbid 1789 | Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfbidf 1790 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |

Theorem | 19.3 1791 | 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.) |

Theorem | 19.9ht 1792 | A closed version of 19.9 1797. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) |

Theorem | 19.9t 1793 | A closed version of 19.9 1797. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.) |

Theorem | 19.9h 1794 | 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.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) |

Theorem | 19.9hOLD 1795 | Obsolete proof of 19.9h 1794 as of 5-Jan-2018. (Contributed by FL, 24-Mar-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.9d 1796 | A deduction version of one direction of 19.9 1797. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |

Theorem | 19.9 1797 | 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.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |

Theorem | 19.9OLD 1798 | Obsolete proof of 19.9 1797 as of 30-Dec-2017. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | hbnt 1799 | Closed theorem version of bound-variable hypothesis builder hbn 1801. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) |

Theorem | hbntOLD 1800 | Obsolete proof of hbnt 1799 as of 3-Mar-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |

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