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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2eximdv 1801* Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1739. (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 → ∃𝑥𝑦𝜒))

Theoremalbidv 1802* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbidv 1803* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theorem2albidv 1804* Formula-building rule for two universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 ↔ ∀𝑥𝑦𝜒))

Theorem2exbidv 1805* Formula-building rule for two existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))

Theorem3exbidv 1806* Formula-building rule for three existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))

Theorem4exbidv 1807* Formula-building rule for four existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))

Theoremalrimiv 1808* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2036 and 19.21v 1821. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremalrimivv 1809* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2036 and 19.21v 1821. (Contributed by NM, 31-Jul-1995.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝑦𝜓)

Theoremalrimdv 1810* Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2036 and 19.21v 1821. (Contributed by NM, 10-Feb-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremexlimiv 1811* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2041.

See exlimi 2043 for a more general version requiring more axioms.

This inference, along with its many variants such as rexlimdv 2916, 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.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 (fictitious) 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 (𝜑𝜓) where 𝜓 is the theorem to be proved and does not contain 𝑥. Then we apply exlimiv 1811 to arrive at (∃𝑥𝜑𝜓). Finally, we separately prove 𝑥𝜑 and detach it with modus ponens ax-mp 5 to arrive at the final theorem 𝜓. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 1838 and ax-8 1940. (Revised by Wolf Lammen, 4-Dec-2017.)

(𝜑𝜓)       (∃𝑥𝜑𝜓)

Theoremexlimiiv 1812* Inference associated with exlimiv 1811. (Contributed by BJ, 19-Dec-2020.)
(𝜑𝜓)    &   𝑥𝜑       𝜓

Theoremexlimivv 1813* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2041. (Contributed by NM, 1-Aug-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑𝜓)

Theoremexlimdv 1814* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2041. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1838, ax-7 1885. (Revised by Wolf Lammen, 4-Dec-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

Theoremexlimdvv 1815* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2041. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓𝜒))

Theoremexlimddv 1816* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theoremnexdv 1817* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremnfdv 1818* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theorem2ax5 1819* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(𝜑 → ∀𝑥𝑦𝜑)

Theoremstdpc5v 1820* Version of stdpc5 2039 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1821. (Revised by Wolf Lammen, 12-Jul-2020.)
(∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem19.21v 1821* Version of 19.21 2036 with a dv condition.

Notational convention: We sometimes suffix with "v" the label of a theorem using a distinct variable ("dv") condition instead of a non-freeness hypothesis such as 𝑥𝜑. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a non-freeness hypothesis ("f" stands for "not free in", see df-nf 1699) instead of a dv condition. For instance, 19.21v 1821 versus 19.21 2036 and vtoclf 3135 versus vtocl 3136. Note that "not free in" is less restrictive than "does not occur in." Note that the version with a dv condition is easily proved from the version with the corresponding non-freeness hypothesis, by using nfv 1796. However, the dv version can often be proved from fewer axioms. (Contributed by NM, 21-Jun-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (Proof shortened by Wolf Lammen, 12-Jul-2020.)

(∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theorem19.32v 1822* Version of 19.32 2096 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Theorem19.31v 1823* Version of 19.31 2097 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

1.4.5  Equality predicate (continued)

The equality predicate was introduced above in wceq 1474 for use by df-tru 1477. See the comments in that section. In this section, we continue with the first "real" use of it.

Theoremweq 1824 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1824 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 1474. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1824 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1474. 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

Theoremequs3 1825 Lemma used in proofs of substitution properties. (Contributed by NM, 10-May-1993.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))

Theoremspeimfw 1826 Specialization, with additional weakening (compared to 19.2 1842) 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-Dec-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

TheoremspeimfwALT 1827 Alternate proof of speimfw 1826 (longer compressed proof, but fewer essential steps). (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

Theoremspimfw 1828 Specialization, with additional weakening (compared to sp 1990) 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, 7-Aug-2017.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑𝜓))

Theoremax12i 1829 Inference that has ax-12 1983 (without 𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 1983 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

1.4.6  Define proper substitution

Syntaxwsb 1830 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

Definitiondf-sb 1831 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." That is, 𝑦 properly replaces 𝑥. For example, [𝑥 / 𝑦]𝑧𝑦 is the same as 𝑧𝑥, as shown in elsb4 2327. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2245.

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 2268, sbcom2 2337 and sbid2v 2349).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2133 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 2347 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 2266. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2322 and sb6 2321.

There are no restrictions on any of the variables, including what variables may occur in wff 𝜑. (Contributed by NM, 10-May-1993.)

([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theoremsbequ2 1832 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))

Theoremsb1 1833 One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2322) or a non-freeness hypothesis (sb5f 2278). (Contributed by NM, 13-May-1993.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremspsbe 1834 A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.)
([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Theoremsbequ8 1835 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))

Theoremsbimi 1836 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)

Theoremsbbii 1837 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)

1.4.7  Axiom scheme ax-6 (Existence)

Axiomax-6 1838 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 axc10 2143 and ax6fromc10 33089. A more convenient form of this axiom is ax6e 2141, 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-6 1838 can be proved from the weaker version ax6v 1839 requiring that the variables be distinct; see theorem ax6 2142.

ax-6 1838 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 ax6vsep 4611.

Except by ax6v 1839, this axiom should not be referenced directly. Instead, use theorem ax6 2142. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦

Theoremax6v 1839* Axiom B7 of [Tarski] p. 75, which requires that 𝑥 and 𝑦 be distinct. This trivial proof is intended merely to weaken axiom ax-6 1838 by adding a distinct variable restriction (\$d). From here on, ax-6 1838 should not be referenced directly by any other proof, so that theorem ax6 2142 will show that we can recover ax-6 1838 from this weaker version if it were an axiom (as it is in the case of Tarski).

Note: Introducing 𝑥, 𝑦 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 prerequisite, similar to adding an unnecessary logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax6v 1839 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-6 1838.

When possible, use of this theorem rather than ax6 2142 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦

Theoremax6ev 1840* At least one individual exists. Weaker version of ax6e 2141. When possible, use of this theorem rather than ax6e 2141 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦

Theoremexiftru 1841 Rule of existential generalization, similar to universal generalization ax-gen 1700, but valid only if an individual exists. Its proof requires ax-6 1838 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1700, ax-4 1713 and this theorem alone, not requiring ax-7 1885 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
𝜑       𝑥𝜑

Theorem19.2 1842 Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic. 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 1999 for a more conventional proof of a more general result, which uses additional axioms. (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 1885. (Revised by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑 → ∃𝑥𝜑)

Theorem19.2d 1843 Deduction associated with 19.2 1842. (Contributed by BJ, 12-May-2019.)
(𝜑 → ∀𝑥𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.8w 1844 Weak version of 19.8a 1988 and instance of 19.2d 1843. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)       (𝜑 → ∃𝑥𝜑)

Theorem19.8v 1845* Version of 19.8a 1988 with a dv condition, requiring fewer axioms. (Contributed by BJ, 12-Mar-2020.)
(𝜑 → ∃𝑥𝜑)

Theorem19.9v 1846* Version of 19.9 2022 with a dv condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1847. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1885. (Revised by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)

Theorem19.3v 1847* Version of 19.3 2018 with a dv condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1846. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1885. (Revised by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑𝜑)

Theoremspvw 1848* Version of sp 1990 when 𝑥 does not occur in 𝜑. Converse of ax-5 1793. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑𝜑)

Theorem19.39 1849 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.24 1850 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.34 1851 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.23v 1852* Version of 19.23 2041 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 28-Jun-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 11-Jan-2020.)
(∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.23vv 1853* Theorem 19.23v 1852 extended to two variables. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theorem19.36v 1854* Version of 19.36 2093 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
(∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.36iv 1855* Inference associated with 19.36v 1854. Version of 19.36i 2094 with a dv condition. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorempm11.53v 1856* Version of pm11.53 2122 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))

Theorem19.12vvv 1857* Version of 19.12vv 2123 with a dv condition, requiring fewer axioms. See also 19.12 2082. (Contributed by BJ, 18-Mar-2020.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))

Theorem19.27v 1858* Version of 19.27 2054 with a dv condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28v 1859* Version of 19.28 2055 with a dv condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.37v 1860* Version of 19.37 2095 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Theorem19.37iv 1861* Inference associated with 19.37v 1860. (Contributed by NM, 5-Aug-1993.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.44v 1862* Version of 19.44 2098 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45v 1863* Version of 19.45 2099 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theorem19.41v 1864* Version of 19.41 2101 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.41vv 1865* Version of 19.41 2101 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theorem19.41vvv 1866* Version of 19.41 2101 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))

Theorem19.41vvvv 1867* Version of 19.41 2101 with four quantifiers and a dv condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))

Theorem19.42v 1868* Version of 19.42 2103 with a dv condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theoremexdistr 1869* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theorem19.42vv 1870* Version of 19.42 2103 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝜓))

Theorem19.42vvv 1871* Version of 19.42 2103 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))

Theoremexdistr2 1872* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))

Theorem3exdistr 1873* Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))

Theorem4exdistr 1874* Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
(∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))

Theoremspimeh 1875* 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 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspimw 1876* 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 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimvw 1877* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspnfw 1878 Weak version of sp 1990. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)
𝜑 → ∀𝑥 ¬ 𝜑)       (∀𝑥𝜑𝜑)

Theoremspfalw 1879 Version of sp 1990 when 𝜑 is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
¬ 𝜑       (∀𝑥𝜑𝜑)

Theoremequs4v 1880* Version of equs4 2181 with a dv condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremequsalvw 1881* Version of equsal 2182 with two dv conditions, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexvw 1882* Version of equsexv 2117 with a dv condition, which requires fewer axioms. See also equsex 2184. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremcbvaliw 1883* 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 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbvalivw 1884* 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 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

1.4.8  Axiom scheme ax-7 (Equality)

Axiomax-7 1885 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. It states that equality is a right-Euclidean binary relation (this is similar, but not identical, to being transitive, which is proved as equtr 1898). 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 1557 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

We prove in ax7 1893 that this axiom can be recovered from its weakened version ax7v 1886 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 1885 should be ax7v 1886. See the comment of ax7v 1886 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 1893 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v 1886* Weakened version of ax-7 1885, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 1885, and it should be referenced only by its two weakened versions ax7v1 1887 and ax7v2 1888, from which ax-7 1885 is then rederived as ax7 1893, which shows that either ax7v 1886 or the conjunction of ax7v1 1887 and ax7v2 1888 is sufficient.

In ax7v 1886, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 1886 "bundles" (a term coined by Raph Levien) its "principal instance" (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧)) with 𝑥, 𝑦, 𝑧 distinct, and its "degenerate instances" (𝑥 = 𝑦 → (𝑥 = 𝑥𝑦 = 𝑥)) and (𝑥 = 𝑦 → (𝑥 = 𝑦𝑦 = 𝑦)) with 𝑥, 𝑦 distinct. These degenerate instances are for instance used in the proofs of equcomiv 1891 and equid 1889 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 1893 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v1 1887* First of two weakened versions of ax7v 1886, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v2 1888* Second of two weakened versions of ax7v 1886, with an extra dv condition on 𝑦, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequid 1889 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, 5-Feb-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
𝑥 = 𝑥

Theoremnfequid 1890 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.)
𝑦 𝑥 = 𝑥

Theoremequcomiv 1891* Weaker form of equcomi 1894 with a dv condition on 𝑥, 𝑦. This is an intermediate step and equcomi 1894 is fully recovered later. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremax6evr 1892* A commuted form of ax6ev 1840. (Contributed by BJ, 7-Dec-2020.)
𝑥 𝑦 = 𝑥

Theoremax7 1893 Proof of ax-7 1885 from ax7v1 1887 and ax7v2 1888, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1886, which is itself a weakened version of ax-7 1885.

Note that the weakened version of ax-7 1885 obtained by adding a dv condition on 𝑥, 𝑧 (resp. on 𝑦, 𝑧) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequcomi 1894 Commutative law for equality. Equality is a symmetric relation. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcom 1895 Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcomd 1896 Deduction form of equcom 1895, symmetry of equality. For the versions for classes, see eqcom 2521 and eqcomd 2520. (Contributed by BJ, 6-Oct-2019.)
(𝜑𝑥 = 𝑦)       (𝜑𝑦 = 𝑥)

Theoremequcoms 1897 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.)
(𝑥 = 𝑦𝜑)       (𝑦 = 𝑥𝜑)

Theoremequtr 1898 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))

Theoremequtrr 1899 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 = 𝑥𝑧 = 𝑦))

Theoremequeuclr 1900 Commuted version of equeucl 1901 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑦 = 𝑥))

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