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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 19.35i 1601 | Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ∃x(φ → ψ) ⇒ ⊢ (∀xφ → ∃xψ) | ||
Theorem | 19.35ri 1602 | Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀xφ → ∃xψ) ⇒ ⊢ ∃x(φ → ψ) | ||
Theorem | 19.25 1603 | Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀y∃x(φ → ψ) → (∃y∀xφ → ∃y∃xψ)) | ||
Theorem | 19.30 1604 | Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (∀x(φ ∨ ψ) → (∀xφ ∨ ∃xψ)) | ||
Theorem | 19.43 1605 | Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) |
⊢ (∃x(φ ∨ ψ) ↔ (∃xφ ∨ ∃xψ)) | ||
Theorem | 19.43OLD 1606 | Obsolete proof of 19.43 1605 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.) |
⊢ (∃x(φ ∨ ψ) ↔ (∃xφ ∨ ∃xψ)) | ||
Theorem | 19.33 1607 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ((∀xφ ∨ ∀xψ) → ∀x(φ ∨ ψ)) | ||
Theorem | 19.33b 1608 | The antecedent provides a condition implying the converse of 19.33 1607. 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.) |
⊢ (¬ (∃xφ ∧ ∃xψ) → (∀x(φ ∨ ψ) ↔ (∀xφ ∨ ∀xψ))) | ||
Theorem | 19.40 1609 | Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ∃xψ)) | ||
Theorem | 19.40-2 1610 | Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃x∃y(φ ∧ ψ) → (∃x∃yφ ∧ ∃x∃yψ)) | ||
Theorem | albiim 1611 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
⊢ (∀x(φ ↔ ψ) ↔ (∀x(φ → ψ) ∧ ∀x(ψ → φ))) | ||
Theorem | 2albiim 1612 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
⊢ (∀x∀y(φ ↔ ψ) ↔ (∀x∀y(φ → ψ) ∧ ∀x∀y(ψ → φ))) | ||
Theorem | exintrbi 1613 | Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.) |
⊢ (∀x(φ → ψ) → (∃xφ ↔ ∃x(φ ∧ ψ))) | ||
Theorem | exintr 1614 | Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.) |
⊢ (∀x(φ → ψ) → (∃xφ → ∃x(φ ∧ ψ))) | ||
Theorem | alsyl 1615 | Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.) |
⊢ ((∀x(φ → ψ) ∧ ∀x(ψ → χ)) → ∀x(φ → χ)) | ||
Axiom | ax-17 1616* |
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 2157 about the logical redundancy of ax-17 1616 in the presence of our obsolete axioms.) This axiom essentially says that if x does not occur in φ, i.e. φ does not depend on x in any way, then we can add the quantifier ∀x to φ with no further assumptions. By sp 1747, we can also remove the quantifier (unconditionally). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) | ||
Theorem | a17d 1617* | ax-17 1616 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.) |
⊢ (φ → (ψ → ∀xψ)) | ||
Theorem | ax17e 1618* | A rephrasing of ax-17 1616 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.) |
⊢ (∃xφ → φ) | ||
Theorem | nfv 1619* | If x is not present in φ, then x is not free in φ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ | ||
Theorem | nfvd 1620* | nfv 1619 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1808. (Contributed by Mario Carneiro, 6-Oct-2016.) |
⊢ (φ → Ⅎxψ) | ||
Theorem | alimdv 1621* | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∀xψ → ∀xχ)) | ||
Theorem | eximdv 1622* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∃xψ → ∃xχ)) | ||
Theorem | 2alimdv 1623* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∀x∀yψ → ∀x∀yχ)) | ||
Theorem | 2eximdv 1624* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∃x∃yψ → ∃x∃yχ)) | ||
Theorem | albidv 1625* | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∀xψ ↔ ∀xχ)) | ||
Theorem | exbidv 1626* | Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃xψ ↔ ∃xχ)) | ||
Theorem | 2albidv 1627* | Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∀x∀yψ ↔ ∀x∀yχ)) | ||
Theorem | 2exbidv 1628* | Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃x∃yψ ↔ ∃x∃yχ)) | ||
Theorem | 3exbidv 1629* | Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃x∃y∃zψ ↔ ∃x∃y∃zχ)) | ||
Theorem | 4exbidv 1630* | Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃x∃y∃z∃wψ ↔ ∃x∃y∃z∃wχ)) | ||
Theorem | alrimiv 1631* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ψ) ⇒ ⊢ (φ → ∀xψ) | ||
Theorem | alrimivv 1632* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
⊢ (φ → ψ) ⇒ ⊢ (φ → ∀x∀yψ) | ||
Theorem | alrimdv 1633* | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (ψ → ∀xχ)) | ||
Theorem | exlimiv 1634* |
Inference from Theorem 19.23 of [Margaris] p.
90.
This inference, along with our many variants such as rexlimdv 2737, 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 A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~hirstjl/primer/hirst.pdf. In informal proofs, the statement "Let C 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 x exists satisfying a wff, i.e. ∃xφ(x) where φ(x) has x free, then we can use φ(C) as a hypothesis for the proof where C 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 φ (containing x) as an antecedent for the main part of the proof. We eventually arrive at (φ → ψ) where ψ is the theorem to be proved and does not contain x. Then we apply exlimiv 1634 to arrive at (∃xφ → ψ). Finally, we separately prove ∃xφ and detach it with modus ponens ax-mp 8 to arrive at the final theorem ψ. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen to remove dependency on ax-9 and ax-8, 4-Dec-2017.) |
⊢ (φ → ψ) ⇒ ⊢ (∃xφ → ψ) | ||
Theorem | exlimivv 1635* | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.) |
⊢ (φ → ψ) ⇒ ⊢ (∃x∃yφ → ψ) | ||
Theorem | exlimdv 1636* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) (Revised by Wolf Lammen to remove dependency on ax-9 and ax-8, 4-Dec-2017.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∃xψ → χ)) | ||
Theorem | exlimdvv 1637* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∃x∃yψ → χ)) | ||
Theorem | exlimddv 1638* | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.) |
⊢ (φ → ∃xψ) & ⊢ ((φ ∧ ψ) → χ) ⇒ ⊢ (φ → χ) | ||
Theorem | nfdv 1639* | Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ (φ → Ⅎxψ) | ||
Theorem | 2ax17 1640* | Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.) |
⊢ (φ → ∀x∀yφ) | ||
Syntax | cv 1641 |
This syntax construction states that a variable x, 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 {y ∣ y ∈ x} is
a class by cab 2339. Since (when
y is distinct from x) we have x = {y ∣ y ∈ x} by
cvjust 2348, we can argue that the syntax "class x " can be viewed as
an abbreviation for "class
{y ∣ y ∈ x}". 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 1641 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1641 is intrinsically no different from any other class-building syntax such as cab 2339, cun 3207, or c0 3550. For a general discussion of the theory of classes and the role of cv 1641, see http://us.metamath.org/mpeuni/mmset.html#class. (The description above applies to set theory, not predicate calculus. The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1643 of predicate calculus from the wceq 1642 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.) |
class x | ||
Syntax | wceq 1642 |
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 wff A = B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1643 of predicate calculus in terms of the wceq 1642 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 x = y could be the = of either weq 1643 or wceq 1642, although mathematically it makes no difference. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2346 for more information on the set theory usage of wceq 1642.) |
wff A = B | ||
Theorem | weq 1643 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1643 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 1642. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1643 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1642. 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.) |
wff x = y | ||
Theorem | equs3 1644 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃x(x = y ∧ φ) ↔ ¬ ∀x(x = y → ¬ φ)) | ||
Theorem | speimfw 1645 | Specialization, with additional weakening to allow bundling of x and y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) |
⊢ (x = y → (φ → ψ)) ⇒ ⊢ (¬ ∀x ¬ x = y → (∀xφ → ∃xψ)) | ||
Theorem | spimfw 1646 | Specialization, with additional weakening to allow bundling of x and y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.) |
⊢ (¬ ψ → ∀x ¬ ψ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (¬ ∀x ¬ x = y → (∀xφ → ψ)) | ||
Theorem | ax11i 1647 | Inference that has ax-11 1746 (without ∀y) 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.) |
⊢ (x = y → (φ ↔ ψ)) & ⊢ (ψ → ∀xψ) ⇒ ⊢ (x = y → (φ → ∀x(x = y → φ))) | ||
Syntax | wsb 1648 | Extend wff definition to include proper substitution (read "the wff that results when y is properly substituted for x in wff φ"). (Contributed by NM, 24-Jan-2006.) |
wff [y / x]φ | ||
Definition | df-sb 1649 |
Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the
preprint). For our notation, we use [y / x]φ to mean "the wff
that results from the proper substitution of y for x in the wff
φ." We can
also use [y / x]φ
in place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 2024.
Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "φ(y) is the wff that results when y is properly substituted for x in φ(x)." For example, if the original φ(x) is x = y, then φ(y) is y = y, from which we obtain that φ(x) is x = x. So what exactly does φ(x) mean? Curry's notation solves this problem. 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 2060, sbcom2 2114 and sbid2v 2123). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1922 shows. We achieve this by having x 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 2119 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 2056. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 2100 and sb6 2099. There are no restrictions on any of the variables, including what variables may occur in wff φ. (Contributed by NM, 5-Aug-1993.) |
⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ))) | ||
Theorem | sbequ2 1650 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → ([y / x]φ → φ)) | ||
Theorem | sb1 1651 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ ([y / x]φ → ∃x(x = y ∧ φ)) | ||
Theorem | sbimi 1652 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
⊢ (φ → ψ) ⇒ ⊢ ([y / x]φ → [y / x]ψ) | ||
Theorem | sbbii 1653 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ ↔ ψ) ⇒ ⊢ ([y / x]φ ↔ [y / x]ψ) | ||
Axiom | ax-9 1654 |
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 x and y 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 1950 and ax9from9o 2148. A more convenient form of this
axiom is a9e 1951, 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 1654 can be proved from the weaker version ax9v 1655 requiring that the variables be distinct; see theorem ax9 1949. ax-9 1654 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 (future). Except by ax9v 1655, this axiom should not be referenced directly. Instead, use theorem ax9 1949. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ ¬ ∀x ¬ x = y | ||
Theorem | ax9v 1655* |
Axiom B7 of [Tarski] p. 75, which requires that
x and y be
distinct. This trivial proof is intended merely to weaken axiom ax-9 1654
by adding a distinct variable restriction. From here on, ax-9 1654
should
not be referenced directly by any other proof, so that theorem ax9 1949
will show that we can recover ax-9 1654 from this weaker version if it were
an axiom (as it is in the case of Tarski).
Note: Introducing xy 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 future theorem that references ax9v 1655 must have a $d specified for the two variables that get substituted for x and y. The $d does not propagate "backwards" i.e. it does not impose a requirement on ax-9 1654. When possible, use of this theorem rather than ax9 1949 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 7-Aug-2015.) |
⊢ ¬ ∀x ¬ x = y | ||
Theorem | a9ev 1656* | At least one individual exists. Weaker version of a9e 1951. When possible, use of this theorem rather than a9e 1951 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 3-Aug-2017.) |
⊢ ∃x x = y | ||
Theorem | exiftru 1657 | A companion rule to ax-gen, valid only if an individual exists. Unlike ax-9 1654, it does not require equality on its interface. Some fundamental theorems of predicate logic can be proven from ax-gen 1546, ax-5 1557 and this theorem alone, not requiring ax-8 1675 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.) |
⊢ φ ⇒ ⊢ ∃xφ | ||
Theorem | exiftruOLD 1658 | Obsolete proof of exiftru 1657 as of 9-Dec-2017. (Contributed by Wolf Lammen, 12-Nov-2017.) (New usage is discouraged.) |
⊢ φ ⇒ ⊢ ∃xφ | ||
Theorem | 19.2 1659 | 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 1757 for a more conventional proof. (Contributed by NM, 2-Aug-2017.) (Revised by Wolf Lammen to remove dependency on ax-8, 4-Dec-2017.) |
⊢ (∀xφ → ∃xφ) | ||
Theorem | 19.8w 1660 | Weak version of 19.8a 1756. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (φ → ∃xφ) | ||
Theorem | 19.39 1661 | Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ((∃xφ → ∃xψ) → ∃x(φ → ψ)) | ||
Theorem | 19.24 1662 | Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ((∀xφ → ∀xψ) → ∃x(φ → ψ)) | ||
Theorem | 19.34 1663 | Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ((∀xφ ∨ ∃xψ) → ∃x(φ ∨ ψ)) | ||
Theorem | 19.9v 1664* | Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.) (Revised by Wolf Lammen to remove dependency on ax-8, 4-Dec-2017.) |
⊢ (∃xφ ↔ φ) | ||
Theorem | 19.3v 1665* | Special case of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 1-Aug-2017.) (Revised by Wolf Lammen to remove dependency on ax-8, 4-Dec-2017.) |
⊢ (∀xφ ↔ φ) | ||
Theorem | spvw 1666* | Version of sp 1747 when x does not occur in φ. This provides the other direction of ax-17 1616. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) |
⊢ (∀xφ → φ) | ||
Theorem | spimeh 1667* | 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.) |
⊢ (φ → ∀xφ) & ⊢ (x = z → (φ → ψ)) ⇒ ⊢ (φ → ∃xψ) | ||
Theorem | spimw 1668* | 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.) |
⊢ (¬ ψ → ∀x ¬ ψ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (∀xφ → ψ) | ||
Theorem | spimvw 1669* | Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (x = y → (φ → ψ)) ⇒ ⊢ (∀xφ → ψ) | ||
Theorem | spnfw 1670 | Weak version of sp 1747. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.) |
⊢ (¬ φ → ∀x ¬ φ) ⇒ ⊢ (∀xφ → φ) | ||
Theorem | sptruw 1671 | Version of sp 1747 when φ is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) |
⊢ φ ⇒ ⊢ (∀xφ → φ) | ||
Theorem | spfalw 1672 | Version of sp 1747 when φ is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.) |
⊢ ¬ φ ⇒ ⊢ (∀xφ → φ) | ||
Theorem | cbvaliw 1673* | Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.) |
⊢ (∀xφ → ∀y∀xφ) & ⊢ (¬ ψ → ∀x ¬ ψ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (∀xφ → ∀yψ) | ||
Theorem | cbvalivw 1674* | Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.) |
⊢ (x = y → (φ → ψ)) ⇒ ⊢ (∀xφ → ∀yψ) | ||
Axiom | ax-8 1675 |
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 1682). 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 x, y, and z 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.) |
⊢ (x = y → (x = z → y = z)) | ||
Theorem | equid 1676 | 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.) |
⊢ x = x | ||
Theorem | equidOLD 1677 | Obsolete proof of equid 1676 as of 9-Dec-2017. (Contributed by NM, 1-Apr-2005.) (New usage is discouraged.) |
⊢ x = x | ||
Theorem | nfequid 1678 | Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.) |
⊢ Ⅎy x = x | ||
Theorem | equcomi 1679 | 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.) |
⊢ (x = y → y = x) | ||
Theorem | equcom 1680 | Commutative law for equality. (Contributed by NM, 20-Aug-1993.) |
⊢ (x = y ↔ y = x) | ||
Theorem | equcoms 1681 | An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → φ) ⇒ ⊢ (y = x → φ) | ||
Theorem | equtr 1682 | A transitive law for equality. (Contributed by NM, 23-Aug-1993.) |
⊢ (x = y → (y = z → x = z)) | ||
Theorem | equtrr 1683 | A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.) |
⊢ (x = y → (z = x → z = y)) | ||
Theorem | equequ1 1684 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) |
⊢ (x = y → (x = z ↔ y = z)) | ||
Theorem | equequ1OLD 1685 | Obsolete version of equequ1 1684 as of 12-Nov-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (x = y → (x = z ↔ y = z)) | ||
Theorem | equequ2 1686 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) |
⊢ (x = y → (z = x ↔ z = y)) | ||
Theorem | stdpc6 1687 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1917.) 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.) |
⊢ ∀x x = x | ||
Theorem | equtr2 1688 | A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ ((x = z ∧ y = z) → x = y) | ||
Theorem | ax12b 1689 | Two equivalent ways of expressing ax-12 1925. See the comment for ax-12 1925. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2017.) |
⊢ ((¬ x = y → (y = z → ∀x y = z)) ↔ (¬ x = y → (¬ x = z → (y = z → ∀x y = z)))) | ||
Theorem | ax12bOLD 1690 | Obsolete version of ax12b 1689 as of 12-Aug-2017. (Contributed by NM, 2-May-2017.) (New usage is discouraged.) |
⊢ ((¬ x = y → (y = z → ∀x y = z)) ↔ (¬ x = y → (¬ x = z → (y = z → ∀x y = z)))) | ||
Theorem | spfw 1691* | Weak version of sp 1747. 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.) |
⊢ (¬ ψ → ∀x ¬ ψ) & ⊢ (∀xφ → ∀y∀xφ) & ⊢ (¬ φ → ∀y ¬ φ) & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∀xφ → φ) | ||
Theorem | spnfwOLD 1692 | Weak version of sp 1747. Uses only Tarski's FOL axiom schemes. Obsolete version of spnfw 1670 as of 13-Aug-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) |
⊢ (¬ φ → ∀x ¬ φ) ⇒ ⊢ (∀xφ → φ) | ||
Theorem | 19.8wOLD 1693 | Obsolete version of 19.8w 1660 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (φ → ∃xφ) | ||
Theorem | spw 1694* | Weak version of specialization scheme sp 1747. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1747 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1747 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1723 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1747 are spfw 1691 (minimal distinct variable requirements), spnfw 1670 (when x is not free in ¬ φ), spvw 1666 (when x does not appear in φ), sptruw 1671 (when φ is true), and spfalw 1672 (when φ is false). (Contributed by NM, 9-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∀xφ → φ) | ||
Theorem | spvwOLD 1695* | Obsolete version of spvw 1666 as of 4-Dec-2017. (Contributed by NM, 10-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∀xφ → φ) | ||
Theorem | 19.3vOLD 1696* | Obsolete version of 19.3v 1665 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∀xφ ↔ φ) | ||
Theorem | 19.9vOLD 1697* | Obsolete version of 19.9v 1664 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.) |
⊢ (∃xφ ↔ φ) | ||
Theorem | exlimivOLD 1698* | Obsolete version of exlimiv 1634 as of 4-Dec-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (φ → ψ) ⇒ ⊢ (∃xφ → ψ) | ||
Theorem | spfalwOLD 1699 | Obsolete proof of spfalw 1672 as of 25-Dec-2017. (Contributed by NM, 23-Apr-1017.) (New usage is discouraged.) |
⊢ ¬ φ ⇒ ⊢ (∀xφ → φ) | ||
Theorem | 19.2OLD 1700 | Obsolete version of 19.2 1659 as of 4-Dec-2017. (Contributed by NM, 2-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∀xφ → ∃xφ) |
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