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

Statement | ||

Theorem | spw 1701* | Weak version of specialization scheme sp 1755. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1755 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1755 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1730 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1755 are spfw 1698 (minimal distinct variable requirements), spnfw 1677 (when is not free in ), spvw 1673 (when does not appear in ), sptruw 1678 (when is true), and spfalw 1679 (when is false). (Contributed by NM, 9-Apr-2017.) |

Theorem | spvwOLD 1702* | Obsolete version of spvw 1673 as of 4-Dec-2017. (Contributed by NM, 10-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.3vOLD 1703* | Obsolete version of 19.3v 1672 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.9vOLD 1704* | Obsolete version of 19.9v 1671 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 1705* | Obsolete version of exlimiv 1641 as of 4-Dec-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | spfalwOLD 1706 | Obsolete proof of spfalw 1679 as of 25-Dec-2017. (Contributed by NM, 23-Apr-1017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.2OLD 1707 | Obsolete version of 19.2 1666 as of 4-Dec-2017. (Contributed by NM, 2-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | cbvalw 1708* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | cbvalvw 1709* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | cbvexvw 1710* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |

Theorem | alcomiw 1711* | Weak version of alcom 1744. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) |

Theorem | hbn1fw 1712* | Weak version of ax-6 1736 from which we can prove any ax-6 1736 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |

Theorem | hbn1w 1713* | Weak version of hbn1 1737. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | hba1w 1714* | Weak version of hba1 1794. See comments for ax6w 1724. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | hbe1w 1715* | Weak version of hbe1 1738. See comments for ax6w 1724. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |

Theorem | hbalw 1716* | Weak version of hbal 1743. Uses only Tarski's FOL axiom schemes. Unlike hbal 1743, 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 1717 |
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 1718 of predicate calculus in terms of the wceq 1649 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 2367 for more information on the set theory usage of wcel 1717.) |

Theorem | wel 1718 |
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 1718 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 1717. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1718 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1717. 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 schemes ax-13 (Left Equality for
Binary Predicate) | ||

Axiom | ax-13 1719 | 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 1720 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |

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

Axiom | ax-14 1721 | 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 1722 | 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 1563, ax-17 1623, ax9v 1662, ax-8 1682, ax-13 1719, and ax-14 1721, together with rule ax-gen 1552. See http://us.metamath.org/mpeuni/mmset.html#compare 1552. 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 1736, ax-7 1741, ax-12 1939, and ax-11 1753, 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 1753 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 2162, 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 ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the and in ax9 1942 are bundled, but they are not in ax9v 1662. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax9v 1662 is the principal instance of ax9 1942. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance of ax9 1942 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($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 1736, ax-7 1741, ax-11 1753, and ax-12 1939 . "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-6 1736, ax-7 1741, ax-11 1753, or ax-12 1939 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 1724, ax7w 1725, ax11w 1728, and ax12w 1731 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-6 1736, ax-7 1741, ax-11 1753, and ax-12 1939 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax6w 1724, ax7w 1725, and ax11w 1728 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 1730 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 1726, ax11dgen 1729, ax12dgen1 1732, ax12dgen2 1733, ax12dgen3 1734, and
ax12dgen4 1735. (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 1661 in an older system, so it seems the main purpose of his later ax9v 1662 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 1661 as our official axiom, we show that the degenerate instance holds in ax9dgen 1723. The case of sp 1755 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 axioms. The best we can do seems to be spw 1701, again requiring substitution instances of that meet conditions (1) and (2) above. Note that our direct proof sp 1755 requires ax-11 1753, which is not part of Tarski's system. | ||

Theorem | ax9dgen 1723 | Tarski's system uses the weaker ax9v 1662 instead of the bundled ax-9 1661, so here we show that the degenerate case of ax-9 1661 can be derived. (Contributed by NM, 23-Apr-2017.) |

Theorem | ax6w 1724* | Weak version of ax-6 1736 from which we can prove any ax-6 1736 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | ax7w 1725* | Weak version of ax-7 1741 from which we can prove any ax-7 1741 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1741, this theorem requires that and be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.) |

Theorem | ax7dgen 1726 | Degenerate instance of ax-7 1741 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax11wlem 1727* | Lemma for weak version of ax-11 1753. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax11w 1728. (Contributed by NM, 10-Apr-2017.) |

Theorem | ax11w 1728* | Weak version of ax-11 1753 from which we can prove any ax-11 1753 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 ). (Contributed by NM, 10-Apr-2017.) |

Theorem | ax11dgen 1729 | Degenerate instance of ax-11 1753 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax11wdemo 1730* | Example of an application of ax11w 1728 that results in an instance of ax-11 1753 for a contrived formula with mixed free and bound variables, , in place of . The proof illustrates bound variable renaming with cbvalvw 1709 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.) |

Theorem | ax12w 1731* | Weak version (principal instance) of ax-12 1939. (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 1724, ax7w 1725, and ax11w 1728. (Contributed by NM, 10-Apr-2017.) |

Theorem | ax12dgen1 1732 | Degenerate instance of ax-12 1939 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax12dgen2 1733 | Degenerate instance of ax-12 1939 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax12dgen3 1734 | Degenerate instance of ax-12 1939 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |

Theorem | ax12dgen4 1735 | Degenerate instance of ax-12 1939 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 1736, ax-7 1741, ax-11 1753, and ax-12 1939 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 1724, ax7w 1725, ax12w 1731, and ax11w 1728, 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 1736 | Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax6w 1724) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.) |

Theorem | hbn1 1737 | is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.) |

Theorem | hbe1 1738 | is not free in . (Contributed by NM, 5-Aug-1993.) |

Theorem | nfe1 1739 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | modal-5 1740 | 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 1741 | 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 1725) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.) |

Theorem | a7s 1742 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |

Theorem | hbal 1743 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) |

Theorem | alcom 1744 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |

Theorem | alrot3 1745 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |

Theorem | alrot4 1746 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |

Theorem | hbald 1747 | Deduction form of bound-variable hypothesis builder hbal 1743. (Contributed by NM, 2-Jan-2002.) |

Theorem | excom 1748 | Theorem 19.11 of [Margaris] p. 89. Revised to remove dependency on ax-11 1753, ax-6 1736, ax-9 1661, ax-8 1682 and ax-17 1623. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 8-Jan-2018.) |

Theorem | excomim 1749 | One direction of Theorem 19.11 of [Margaris] p. 89. Revised to remove dependency on ax-11 1753, ax-6 1736, ax-9 1661, ax-8 1682 and ax-17 1623. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Revised by Wolf Lammen, 8-Jan-2018.) |

Theorem | excom13 1750 | Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |

Theorem | exrot3 1751 | Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) |

Theorem | exrot4 1752 | Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) |

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

Axiom | ax-11 1753 |
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 2125).
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 2168 ("o" for "old") and was replaced with this shorter ax-11 1753 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 2020. Conversely, this axiom is proved from ax-11o 2168 as theorem ax11 2182. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2168) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. See ax11v 2122 and ax11v2 2018 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax11w 1728) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.) |

Theorem | 19.8a 1754 | 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 1755 |
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 2050. This theorem shows that our obsolete axiom ax-4 2162 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 1753. It is thought the best we can do using only Tarski's axioms is spw 1701. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) |

Theorem | spOLD 1756 | Obsolete proof of sp 1755 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 1757 |
Show that the original axiom ax-5o 2163 can be derived from ax-5 1563
and
others. See ax5 2173 for the rederivation of ax-5 1563
from ax-5o 2163.
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 1758 |
Show that the original axiom ax-6o 2164 can be derived from ax-6 1736
and
others. See ax6 2174 for the rederivation of ax-6 1736
from ax-6o 2164.
Normally, ax6o 1758 should be used rather than ax-6o 2164, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.) |

Theorem | a6e 1759 | Abbreviated version of ax6o 1758. (Contributed by NM, 5-Aug-1993.) |

Theorem | modal-b 1760 | The analog in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.) |

Theorem | spi 1761 | Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.) |

Theorem | sps 1762 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |

Theorem | spsd 1763 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |

Theorem | 19.8aOLD 1764 | 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 1765 | Theorem 19.2 of [Margaris] p. 89, generalized to use two set variables. (Contributed by O'Cat, 31-Mar-2008.) |

Theorem | 19.21bi 1766 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.23bi 1767 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | nexr 1768 | Inference from 19.8a 1754. (Contributed by Jeff Hankins, 26-Jul-2009.) |

Theorem | nfr 1769 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |

Theorem | nfri 1770 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfrd 1771 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | alimd 1772 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | alrimi 1773 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfd 1774 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfdh 1775 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | alrimdd 1776 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | alrimd 1777 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | eximd 1778 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nexd 1779 | Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | albid 1780 | Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | exbid 1781 | Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfbidf 1782 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |

Theorem | hbnt 1783 | Closed theorem version of bound-variable hypothesis builder hbn 1784. (Contributed by NM, 5-Aug-1993.) |

Theorem | hbn 1784 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.) |

Theorem | hbnOLD 1785 | Obsolete proof of hbn 1784 as of 16-Dec-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.9ht 1786 | A closed version of 19.9 1791. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.9t 1787 | A closed version of 19.9 1791. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.) |

Theorem | 19.9h 1788 | 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 1789 | Obsolete proof of 19.9h 1788 as of 5-Jan-2018. (Contributed by FL, 24-Mar-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | 19.9d 1790 | A deduction version of one direction of 19.9 1791. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |

Theorem | 19.9 1791 | 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 1792 | Obsolete proof of 19.9 1791 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 | 19.3 1793 | 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 | hba1 1794 | 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.) (Proof shortened by Wolf Lammen, 15-Dec-2017.) |

Theorem | hba1OLD 1795 | Obsolete proof of hba1 1794 as of 15-Dec-2017 (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | nfa1 1796 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | a5i 1797 | Inference version of ax5o 1757. (Contributed by NM, 5-Aug-1993.) |

Theorem | nfnf1 1798 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfnd 1799 | If in a context is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) |

Theorem | nfndOLD 1800 | Obsolete proof of nfnd 1799 as of 28-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |

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