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

Statement | ||

Theorem | 19.29 1601 | Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |

Theorem | 19.29r 1602 | Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |

Theorem | 19.29r2 1603 | Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.) |

Theorem | 19.29x 1604 | Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.) |

Theorem | 19.35 1605 | Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) |

Theorem | 19.35i 1606 | Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.35ri 1607 | Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.25 1608 | Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.30 1609 | Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |

Theorem | 19.43 1610 | Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) |

Theorem | 19.43OLD 1611 | Obsolete proof of 19.43 1610 as of 3-May-2016. Leave this in for the example on the mmrecent.html page. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | 19.33 1612 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.33b 1613 | The antecedent provides a condition implying the converse of 19.33 1612. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.) |

Theorem | 19.40 1614 | Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.40-2 1615 | Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |

Theorem | albiim 1616 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |

Theorem | 2albiim 1617 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |

Theorem | exintrbi 1618 | Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.) |

Theorem | exintr 1619 | Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.) |

Theorem | alsyl 1620 | Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.) |

1.4.4 Axiom scheme ax-17 (Distinctness) - first
use of $d | ||

Axiom | ax-17 1621* |
Axiom of Distinctness. This axiom quantifies a variable over a formula
in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the
preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski]
p. 77 and Axiom C5-1 of [Monk2] p. 113.
(See comments in ax17o 2170 about the logical redundancy of ax-17 1621 in the presence of our obsolete axioms.) This axiom essentially says that if does not occur in , i.e. does not depend on in any way, then we can add the quantifier to with no further assumptions. By sp 1753, we can also remove the quantifier (unconditionally). (Contributed by NM, 5-Aug-1993.) |

Theorem | a17d 1622* | ax-17 1621 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.) |

Theorem | ax17e 1623* | A rephrasing of ax-17 1621 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.) |

Theorem | nfv 1624* | If is not present in , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfvd 1625* | nfv 1624 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1815. (Contributed by Mario Carneiro, 6-Oct-2016.) |

Theorem | alimdv 1626* | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.) |

Theorem | eximdv 1627* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |

Theorem | 2alimdv 1628* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.) |

Theorem | 2eximdv 1629* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.) |

Theorem | albidv 1630* | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |

Theorem | exbidv 1631* | Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |

Theorem | 2albidv 1632* | Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.) |

Theorem | 2exbidv 1633* | Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.) |

Theorem | 3exbidv 1634* | Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.) |

Theorem | 4exbidv 1635* | Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.) |

Theorem | alrimiv 1636* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | alrimivv 1637* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |

Theorem | alrimdv 1638* | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) |

Theorem | exlimiv 1639* |
Inference from Theorem 19.23 of [Margaris] p.
90.
This inference, along with our many variants such as rexlimdv 2751, is
used to implement a metatheorem called "Rule C" that is given
in many
logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C
in [Margaris] p. 40, or Rule C in
Hirst and Hirst's In informal proofs, the statement "Let be an element such that..." almost always means an implicit application of Rule C. In essence, Rule C states that if we can prove that some element exists satisfying a wff, i.e. where has free, then we can use as a hypothesis for the proof where is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier. We cannot do this in Metamath directly. Instead, we use the original (containing ) as an antecedent for the main part of the proof. We eventually arrive at where is the theorem to be proved and does not contain . Then we apply exlimiv 1639 to arrive at . Finally, we separately prove and detach it with modus ponens ax-mp 8 to arrive at the final theorem . (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen to remove dependency on ax-9 and ax-8, 4-Dec-2017.) |

Theorem | exlimivv 1640* | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.) |

Theorem | exlimdv 1641* | Deduction from Theorem 19.23 of [Margaris] p. 90. Revised to remove dependency on ax-9 1659 and ax-8 1680. (Contributed by NM, 27-Apr-1994.) (Revised by Wolf Lammen, 4-Dec-2017.) |

Theorem | exlimdvv 1642* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |

Theorem | exlimddv 1643* | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.) |

Theorem | nfdv 1644* | Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | 2ax17 1645* | Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.) |

1.4.5 Equality predicate; define
substitution | ||

Syntax | cv 1646 |
This syntax construction states that a variable , which has been
declared to be a set variable by $f statement vx, is also a class
expression. This can be justified informally as follows. We know that
the class builder is a class by cab 2352.
Since (when
is distinct from
) we have by
cvjust 2361, we can argue that the syntax " " can be viewed as
an abbreviation for " ". See the
discussion
under the definition of class in [Jech] p. 4
showing that "Every set can
be considered to be a class."
While it is tempting and perhaps occasionally useful to view cv 1646 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1646 is intrinsically no different from any other class-building syntax such as cab 2352, cun 3236, or c0 3543. For a general discussion of the theory of classes and the role of cv 1646, see http://us.metamath.org/mpeuni/mmset.html#class. (The description above applies to set theory, not predicate calculus. The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1648 of predicate calculus from the wceq 1647 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.) |

Syntax | wceq 1647 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1648 of predicate calculus in terms of the wceq 1647 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. For example, some parsers - although not the Metamath program - stumble on the fact that the in could be the of either weq 1648 or wceq 1647, although mathematically it makes no difference. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2359 for more information on the set theory usage of wceq 1647.) |

Theorem | weq 1648 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1648 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 wceq 1647. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1648 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1647. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |

Theorem | equs3 1649 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) |

Theorem | speimfw 1650 | Specialization, with additional weakening to allow bundling of and . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) |

Theorem | spimfw 1651 | Specialization, with additional weakening to allow bundling of and . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.) |

Theorem | ax11i 1652 | Inference that has ax-11 1751 (without ) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.) |

Syntax | wsb 1653 | Extend wff definition to include proper substitution (read "the wff that results when is properly substituted for in wff "). (Contributed by NM, 24-Jan-2006.) |

Definition | df-sb 1654 |
Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the
preprint). For our notation, we use to mean "the wff
that results from the proper substitution of for in the wff
." We
can also use in
place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 2037.
Our notation was introduced in Haskell B. Curry's In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2073, sbcom2 2127 and sbid2v 2136). Note that our definition is valid even when and are replaced with the same variable, as sbid 1934 shows. We achieve this by having free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2132 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2069. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2113 and sb6 2112. There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.) |

Theorem | sbequ2 1655 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |

Theorem | sb1 1656 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |

Theorem | sbimi 1657 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |

Theorem | sbbii 1658 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |

1.4.6 Axiom scheme ax-9 (Existence) | ||

Axiom | ax-9 1659 |
Axiom of Existence. One of the equality and substitution axioms of
predicate calculus with equality. This axiom tells us is that at least
one thing exists. In this form (not requiring that and be
distinct) it was used in an axiom system of Tarski (see Axiom B7' in
footnote 1 of [KalishMontague] p.
81.) It is equivalent to axiom scheme
C10' in [Megill] p. 448 (p. 16 of the
preprint); the equivalence is
established by ax9o 1963 and ax9from9o 2161. A more convenient form of this
axiom is a9e 1964, which has additional remarks.
Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html. ax-9 1659 can be proved from the weaker version ax9v 1660 requiring that the variables be distinct; see theorem ax9 1962. ax-9 1659 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax9vsep 4247. Except by ax9v 1660, this axiom should not be referenced directly. Instead, use theorem ax9 1962. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |

Theorem | ax9v 1660* |
Axiom B7 of [Tarski] p. 75, which requires that
and be
distinct. This trivial proof is intended merely to weaken axiom ax-9 1659
by adding a distinct variable restriction. From here on, ax-9 1659
should
not be referenced directly by any other proof, so that theorem ax9 1962
will show that we can recover ax-9 1659 from this weaker version if it were
an axiom (as it is in the case of Tarski).
Note: Introducing as a distinct
variable group "out of the
blue" with no apparent justification has puzzled some people, but
it is
perfectly sound. All we are doing is adding an additional redundant
requirement, no different from adding a redundant logical hypothesis,
that results in a weakening of the theorem. This means that any
When possible, use of this theorem rather than ax9 1962 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 7-Aug-2015.) |

Theorem | a9ev 1661* | At least one individual exists. Weaker version of a9e 1964. When possible, use of this theorem rather than a9e 1964 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 3-Aug-2017.) |

Theorem | exiftru 1662 | A companion rule to ax-gen, valid only if an individual exists. Unlike ax-9 1659, it does not require equality on its interface. Some fundamental theorems of predicate logic can be proven from ax-gen 1551, ax-5 1562 and this theorem alone, not requiring ax-8 1680 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.) |

Theorem | exiftruOLD 1663 | Obsolete proof of exiftru 1662 as of 9-Dec-2017. (Contributed by Wolf Lammen, 12-Nov-2017.) (New usage is discouraged.) |

Theorem | 19.2 1664 | Theorem 19.2 of [Margaris] p. 89. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 1763 for a more conventional proof. Revised to remove dependency on ax-8 1680. (Contributed by NM, 2-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.) |

Theorem | 19.8w 1665 | Weak version of 19.8a 1752. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) |

Theorem | 19.39 1666 | Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.24 1667 | Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.34 1668 | Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.9v 1669* | Special case of Theorem 19.9 of [Margaris] p. 89. Revised to remove dependency on ax-8 1680. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.) |

Theorem | 19.3v 1670* | Special case of Theorem 19.3 of [Margaris] p. 89. Revised to remove dependency on ax-8 1680. (Contributed by NM, 1-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.) |

Theorem | spvw 1671* | Version of sp 1753 when does not occur in . This provides the other direction of ax-17 1621. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) |

Theorem | spimeh 1672* | Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) |

Theorem | spimw 1673* | Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.) |

Theorem | spimvw 1674* | Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |

Theorem | spnfw 1675 | Weak version of sp 1753. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.) |

Theorem | sptruw 1676 | Version of sp 1753 when is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) |

Theorem | spfalw 1677 | Version of sp 1753 when is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.) |

Theorem | cbvaliw 1678* | Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.) |

Theorem | cbvalivw 1679* | Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.) |

1.4.7 Axiom scheme ax-8 (Equality) | ||

Axiom | ax-8 1680 |
Axiom of Equality. One of the equality and substitution axioms of
predicate calculus with equality. This is similar to, but not quite, a
transitive law for equality (proved later as equtr 1687). 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 C7 of [Monk2] p. 105 and Axiom Scheme
C8' in [Megill] p. 448 (p. 16
of the preprint).
The equality symbol was invented in 1527 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle." Note that this axiom is still valid even when any two or all three of , , and are replaced with the same variable since they do not have any distinct variable (Metamath's $d) restrictions. Because of this, we say that these three variables are "bundled" (a term coined by Raph Levien). (Contributed by NM, 5-Aug-1993.) |

Theorem | equid 1681 | Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.) |

Theorem | equidOLD 1682 | Obsolete proof of equid 1681 as of 9-Dec-2017. (Contributed by NM, 1-Apr-2005.) (New usage is discouraged.) |

Theorem | nfequid 1683 | Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.) |

Theorem | equcomi 1684 | Commutative law for equality. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 9-Apr-2017.) |

Theorem | equcom 1685 | Commutative law for equality. (Contributed by NM, 20-Aug-1993.) |

Theorem | equcoms 1686 | An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.) |

Theorem | equtr 1687 | A transitive law for equality. (Contributed by NM, 23-Aug-1993.) |

Theorem | equtrr 1688 | A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.) |

Theorem | equequ1 1689 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) |

Theorem | equequ1OLD 1690 | Obsolete version of equequ1 1689 as of 12-Nov-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | equequ2 1691 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) |

Theorem | stdpc6 1692 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1929.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.) |

Theorem | equtr2 1693 | A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |

Theorem | ax12b 1694 | Two equivalent ways of expressing ax-12 1937. See the comment for ax-12 1937. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2017.) |

Theorem | ax12bOLD 1695 | Obsolete version of ax12b 1694 as of 12-Aug-2017. (Contributed by NM, 2-May-2017.) (New usage is discouraged.) |

Theorem | spfw 1696* | Weak version of sp 1753. 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 1697 | Weak version of sp 1753. Uses only Tarski's FOL axiom schemes. Obsolete version of spnfw 1675 as of 13-Aug-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) |

Theorem | 19.8wOLD 1698 | Obsolete version of 19.8w 1665 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | spw 1699* | Weak version of specialization scheme sp 1753. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1753 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1753 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1728 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1753 are spfw 1696 (minimal distinct variable requirements), spnfw 1675 (when is not free in ), spvw 1671 (when does not appear in ), sptruw 1676 (when is true), and spfalw 1677 (when is false). (Contributed by NM, 9-Apr-2017.) |

Theorem | spvwOLD 1700* | Obsolete version of spvw 1671 as of 4-Dec-2017. (Contributed by NM, 10-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |

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