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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | equeucl 1901 | Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 1885.) Exported (curried) form of equtr2 1904. (Contributed by BJ, 11-Apr-2021.) |
⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | ||
Theorem | equequ1 1902 | An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | ||
Theorem | equequ2 1903 | An equivalence law for equality. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) (Proof shortened by BJ, 12-Apr-2021.) |
⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦)) | ||
Theorem | equtr2 1904 | Equality is a left-Euclidean binary relation. Imported (uncurried) form of equeucl 1901. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) | ||
Theorem | equequ2OLD 1905 | Obsolete proof of equequ2 1903 as of 12-Apr-2021. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦)) | ||
Theorem | equtr2OLD 1906 | Obsolete proof of equtr2 1904 as of 11-Apr-2021. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) | ||
Theorem | stdpc6 1907 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1908.) 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 | stdpc7 1908 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1907.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑥)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑)) | ||
Theorem | equvinv 1909* | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1966, ax-13 2137. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) |
⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦)) | ||
Theorem | equviniva 1910* | A modified version of the forward implication of equvinv 1909 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.) |
⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) | ||
Theorem | equvinivOLD 1911* | The forward implication of equvinv 1909. Obsolete as of 11-Apr-2021. Use equvinv 1909 instead. (Contributed by Wolf Lammen, 11-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦)) | ||
Theorem | equvinvOLD 1912* | Obsolete version of equvinv 1909 as of 11-Apr-2021. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1966, ax-13 2137. (Revised by Wolf Lammen, 10-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) | ||
Theorem | equvelv 1913* | A specialized version of equvel 2239 with distinct variable restrictions and fewer axiom usage. (Contributed by Wolf Lammen, 10-Apr-2021.) |
⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦)) | ||
Theorem | ax13b 1914 | An equivalence between two ways of expressing ax-13 2137. See the comment for ax-13 2137. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.) |
⊢ ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝜑)))) | ||
Theorem | spfw 1915* | Weak version of sp 1990. 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. (Contributed by NM, 19-Apr-2017.) |
⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) & ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) & ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | spw 1916* | Weak version of the specialization scheme sp 1990. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1990 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1990 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1960 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1990 are spfw 1915 (minimal distinct variable requirements), spnfw 1878 (when 𝑥 is not free in ¬ 𝜑), spvw 1848 (when 𝑥 does not appear in 𝜑), sptruw 1712 (when 𝜑 is true), and spfalw 1879 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | cbvalw 1917* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) & ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) & ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) & ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbvalvw 1918* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbvexvw 1919* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Theorem | alcomiw 1920* | Weak version of alcom 1974. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | hbn1fw 1921* | Weak version of ax-10 1966 from which we can prove any ax-10 1966 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.) |
⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) & ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) & ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) & ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) & ⊢ (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hbn1w 1922* | Weak version of hbn1 1967. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hba1w 1923* | Weak version of hba1 2026. See comments for ax10w 1954. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | hbe1w 1924* | Weak version of hbe1 1968. See comments for ax10w 1954. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | ||
Theorem | hbalw 1925* | Weak version of hbal 1973. Uses only Tarski's FOL axiom schemes. Unlike hbal 1973, this theorem requires that 𝑥 and 𝑦 be distinct, i.e. not be bundled. (Contributed by NM, 19-Apr-2017.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) | ||
Theorem | spaev 1926* |
A special instance of sp 1990 applied to an equality with a dv condition.
Unlike the more general sp 1990, we can prove this without ax-12 1983.
Instance of aeveq 1930.
The antecedent ∀𝑥𝑥 = 𝑦 with distinct 𝑥 and 𝑦 is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term. Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition ∀𝑥𝑥 = 𝑦 is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | ||
Theorem | cbvaev 1927* | Change bound variable in an equality with a dv condition. Instance of aev 1931. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦) | ||
Theorem | aevlem0 1928* | Lemma for aevlem 1929. Instance of aev 1931. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 1983. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 30-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥) | ||
Theorem | aevlem 1929* | Lemma for aev 1931 and axc16g 2071. Change free and bound variables. Instance of aev 1931. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2137, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡) | ||
Theorem | aeveq 1930* | The antecedent ∀𝑥𝑥 = 𝑦 with a dv condition (typical of a one-object universe) forces equality of everything. (Contributed by Wolf Lammen, 19-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝑧 = 𝑡) | ||
Theorem | aev 1931* | A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 1971. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2137, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 1983. (Revised by Wolf Lammen, 19-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢) | ||
Theorem | hbaevg 1932* | Generalization of hbaev 1933, proved at no extra cost. Instance of aev2 1934. (Contributed by Wolf Lammen, 22-Mar-2021.) (Revised by BJ, 29-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑡 = 𝑢) | ||
Theorem | hbaev 1933* | Version of hbae 2207 with a DV condition, requiring fewer axioms. Instance of hbaevg 1932 and aev2 1934. (Contributed by Wolf Lammen, 22-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦) | ||
Theorem | aev2 1934* |
A version of aev 1931 with two universal quantifiers in the
consequent, and
a generalization of hbaevg 1932. One can prove similar statements with
arbitrary numbers of universal quantifiers in the consequent (the series
begins with aeveq 1930, aev 1931, aev2 1934).
Using aev 1931 and alrimiv 1808 (as in aev2ALT 1935), one can actually prove (with no more axioms) any scheme of the form (∀𝑥𝑥 = 𝑦 → PHI) , DV (𝑥, 𝑦) where PHI involves only setvar variables and the connectors →, ↔, ∧, ∨, ⊤, =, ∀, ∃, ∃*, ∃!, Ⅎ. An example is given by aevdemo 26441. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1700, ax-1 6-- ax-13 2137 (as the one-element universe shows). (Contributed by BJ, 29-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣) | ||
Theorem | aev2ALT 1935* | Alternate proof of aev2 1934, bypassing hbaevg 1932. (Contributed by BJ, 23-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣) | ||
Theorem | axc11nlemOLD2 1936* | Lemma for axc11n 2199. Change bound variable in an equality. Obsolete as of 29-Mar-2021. Use aev 1931 instead. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Restructure to ease either bundling, or reducing dependencies on axioms. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 1983. (Revised by Wolf Lammen, 14-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | aevlemOLD 1937* | Old proof of aevlem 1929. Obsolete as of 29-Mar-2021. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2137, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑧 𝑧 = 𝑤 → ∀𝑦 𝑦 = 𝑥) | ||
Syntax | wcel 1938 |
Extend wff definition to include the membership connective between
classes.
For a general discussion of the theory of classes, see mmset.html#class. (The purpose of introducing wff 𝐴 ∈ 𝐵 here is to allow us to express i.e. "prove" the wel 1939 of predicate calculus in terms of the wcel 1938 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 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2501 for more information on the set theory usage of wcel 1938.) |
wff 𝐴 ∈ 𝐵 | ||
Theorem | wel 1939 |
Extend wff definition to include atomic formulas with the epsilon
(membership) predicate. This is read "𝑥 is an element of
𝑦," "𝑥 is a member of 𝑦,"
"𝑥 belongs to 𝑦,"
or "𝑦 contains 𝑥." Note: The
phrase "𝑦 includes
𝑥 " means "𝑥 is a
subset of 𝑦;" to use it also for
𝑥
∈ 𝑦, as some
authors occasionally do, is poor form and causes
confusion, according to George Boolos (1992 lecture at MIT).
This syntactic 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 1939 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 1938. This lets us avoid overloading the ∈ connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1939 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1938. 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.) |
wff 𝑥 ∈ 𝑦 | ||
Axiom | ax-8 1940 |
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.
We prove in ax8 1944 that this axiom can be recovered from its weakened version ax8v 1941 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-8 1940 should be ax8v 1941. See the comment of ax8v 1941 for more details on these matters. (Contributed by NM, 30-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax8 1944 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | ax8v 1941* | Weakened version of ax-8 1940, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-8 1940, and it should be referenced only by its two weakened versions ax8v1 1942 and ax8v2 1943, from which ax-8 1940 is then rederived as ax8 1944, which shows that either ax8v 1941 or the conjunction of ax8v1 1942 and ax8v2 1943 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax8 1944 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | ax8v1 1942* | First of two weakened versions of ax8v 1941, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | ax8v2 1943* | Second of two weakened versions of ax8v 1941, with an extra dv condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | ax8 1944 | Proof of ax-8 1940 from ax8v1 1942 and ax8v2 1943, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 1941, which is itself a weakened version of ax-8 1940. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | elequ1 1945 | An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | ||
Theorem | cleljust 1946* | When the class variables in definition df-clel 2510 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1939 with the class variables in wcel 1938. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 1882 in order to remove dependencies on ax-10 1966, ax-12 1983, ax-13 2137. Note that there is no DV condition on 𝑥, 𝑦, that is, on the variables of the left-hand side. (Revised by BJ, 29-Dec-2020.) |
⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) | ||
Axiom | ax-9 1947 |
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).
We prove in ax9 1951 that this axiom can be recovered from its weakened version ax9v 1948 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-9 1947 should be ax9v 1948. See the comment of ax9v 1948 for more details on these matters. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax9 1951 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | ax9v 1948* | Weakened version of ax-9 1947, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-9 1947, and it should be referenced only by its two weakened versions ax9v1 1949 and ax9v2 1950, from which ax-9 1947 is then rederived as ax9 1951, which shows that either ax9v 1948 or the conjunction of ax9v1 1949 and ax9v2 1950 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax9 1951 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | ax9v1 1949* | First of two weakened versions of ax9v 1948, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | ax9v2 1950* | Second of two weakened versions of ax9v 1948, with an extra dv condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | ax9 1951 | Proof of ax-9 1947 from ax9v1 1949 and ax9v2 1950, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 1948, which is itself a weakened version of ax-9 1947. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | elequ2 1952 | An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | ||
The original axiom schemes of Tarski's predicate calculus are ax-4 1713, ax-5 1793, ax6v 1839, ax-7 1885, ax-8 1940, and ax-9 1947, together with rule ax-gen 1700. See mmset.html#compare 1700. 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-10 1966, ax-11 1971, ax-12 1983, and ax-13 2137, which are not part of Tarski's axiom schemes. Each object-language instance of them is provable from Tarski's axioms, so they are logically redundant. However, they are conjectured not to be provable directly as schemes from Tarski's axiom schemes using only Metamath's direct substitution rule. They are used to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and setvar variables, bundled or not, whose object-language instances are valid. (ax-12 1983 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-c5 33070, but they can all be proved as theorems from the above.) Terminology: Two setvar (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the 𝑥 and 𝑦 in ax-6 1838 are bundled, but they are not in ax6v 1839. We also say that a scheme is bundled when it has at least one pair of bundled setvar variables. If distinct variable conditions are added to all setvar variable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1839 is the principal instance of ax-6 1838. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance ¬ ∀𝑥¬ 𝑥 = 𝑥 of ax-6 1838 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with, and theorems are more general. There may be some economy in being able to prove facts about 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-10 1966, ax-11 1971, ax-12 1983, and ax-13 2137. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 1966, ax-11 1971, ax-12 1983, or ax-13 2137 in which (1) there are no wff metavariables and (2) all setvar variables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each setvar variable 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 theorem schemes ax10w 1954, ax11w 1955, ax12w 1958, and ax13w 1961 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 1966, ax-11 1971, ax-12 1983, and ax-13 2137 meeting Conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 1954, ax11w 1955, and ax12w 1958 is of the form (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) where 𝜓 is an auxiliary or "dummy" wff metavariable in which 𝑥 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 ax12wdemo 1960 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 ax11dgen 1956, ax12dgen 1959, ax13dgen1 1962, ax13dgen2 1963, ax13dgen3 1964, and ax13dgen4 1965. (Their proofs are trivial, but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-10 1966, ax-11 1971, ax-12 1983, and ax-13 2137 are schemes of Tarski's system, meaning that all object-language instances they generate are theorems of Tarski's system. It is interesting that Tarski used the bundled scheme ax-6 1838 in an older system, so it seems the main purpose of his later ax6v 1839 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-6 1838 as our official axiom, we show that the degenerate instance holds in ax6dgen 1953. (Recall that in set.mm, the only statement referencing ax-6 1838 is ax6v 1839.) The case of sp 1990 is curious: originally an axiom scheme of Tarski's system, it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the scheme form ∀𝑥𝜑 → 𝜑 apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 1916, again requiring substitution instances of 𝜑 that meet Conditions (1) and (2) above. Note that our direct proof sp 1990 requires ax-12 1983, which is not part of Tarski's system. | ||
Theorem | ax6dgen 1953 | Tarski's system uses the weaker ax6v 1839 instead of the bundled ax-6 1838, so here we show that the degenerate case of ax-6 1838 can be derived. Even though ax-6 1838 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1838 is ax6v 1839. We later rederive from ax6v 1839 the bundled form as ax6 2142 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥 | ||
Theorem | ax10w 1954* | Weak version of ax-10 1966 from which we can prove any ax-10 1966 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | ax11w 1955* | Weak version of ax-11 1971 from which we can prove any ax-11 1971 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 1971, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | ax11dgen 1956 | Degenerate instance of ax-11 1971 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (∀𝑥∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | ax12wlem 1957* | Lemma for weak version of ax-12 1983. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 1958. (Contributed by NM, 10-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax12w 1958* | Weak version of ax-12 1983 from which we can prove any ax-12 1983 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that 𝑥 and 𝑦 be distinct (unless 𝑥 does not occur in 𝜑). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for 𝜑, see ax12wdemo 1960. (Contributed by NM, 10-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax12dgen 1959 | Degenerate instance of ax-12 1983 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (𝑥 = 𝑥 → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑥 → 𝜑))) | ||
Theorem | ax12wdemo 1960* | Example of an application of ax12w 1958 that results in an instance of ax-12 1983 for a contrived formula with mixed free and bound variables, (𝑥 ∈ 𝑦 ∧ ∀𝑥𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧𝑦 ∈ 𝑥), in place of 𝜑. The proof illustrates bound variable renaming with cbvalvw 1918 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (∀𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥)))) | ||
Theorem | ax13w 1961* | Weak version (principal instance) of ax-13 2137. (Because 𝑦 and 𝑧 don't need to be distinct, this actually bundles the principal instance and the degenerate instance (¬ 𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥𝑦 = 𝑦)).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 1954, ax11w 1955, and ax12w 1958. (Contributed by NM, 10-Apr-2017.) |
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
Theorem | ax13dgen1 1962 | Degenerate instance of ax-13 2137 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (¬ 𝑥 = 𝑥 → (𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧)) | ||
Theorem | ax13dgen2 1963 | Degenerate instance of ax-13 2137 where bundled variables 𝑥 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥)) | ||
Theorem | ax13dgen3 1964 | Degenerate instance of ax-13 2137 where bundled variables 𝑦 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥 𝑦 = 𝑦)) | ||
Theorem | ax13dgen4 1965 | Degenerate instance of ax-13 2137 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes . (Contributed by NM, 13-Apr-2017.) |
⊢ (¬ 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) | ||
In this section we introduce four additional schemes ax-10 1966, ax-11 1971, ax-12 1983, and ax-13 2137 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 "scheme completeness," which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs, and setvar variables 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 ax10w 1954, ax11w 1955, ax12w 1958, and ax13w 1961, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all setvar variables 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 metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 1983 from all others has been shown, and independence of Tarski's ax-6 1838 from all others has been shown; see items 9a and 11 on http://us.metamath.org/award2003.html. | ||
Axiom | ax-10 1966 | Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 1954) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 21-May-2008.) Use its alias hbn1 1967 instead if you must use it. Any theorem in first order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 1966 through ax-13 2137, by invoking ax10w 1954 through ax13w 1961. We encourage proving theorems *without* ax-10 1966 through ax-13 2137 and moving them up to the ax-4 1713 through ax-9 1947 section. (New usage is discouraged.) |
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hbn1 1967 | Alias for ax-10 1966 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.) |
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hbe1 1968 | The setvar 𝑥 is not free in ∃𝑥𝜑. (Contributed by NM, 24-Jan-1993.) |
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | ||
Theorem | nfe1 1969 | The setvar 𝑥 is not free in ∃𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥∃𝑥𝜑 | ||
Theorem | modal-5 1970 | The analogue in our predicate calculus of axiom (5) of modal logic S5. (Contributed by NM, 5-Oct-2005.) |
⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑) | ||
Axiom | ax-11 1971 | 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 ax11w 1955) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | alcoms 1972 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
⊢ (∀𝑥∀𝑦𝜑 → 𝜓) ⇒ ⊢ (∀𝑦∀𝑥𝜑 → 𝜓) | ||
Theorem | hbal 1973 | If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 12-Mar-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) | ||
Theorem | alcom 1974 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.) |
⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑) | ||
Theorem | alrot3 1975 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑦∀𝑧∀𝑥𝜑) | ||
Theorem | alrot4 1976 | Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |
⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑) | ||
Theorem | hbald 1977 | Deduction form of bound-variable hypothesis builder hbal 1973. (Contributed by NM, 2-Jan-2002.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)) | ||
Theorem | excom 1978 | Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-5 1793, ax-6 1838, ax-7 1885, ax-10 1966, ax-12 1983. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.) |
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) | ||
Theorem | excomim 1979 | One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-5 1793, ax-6 1838, ax-7 1885, ax-10 1966, ax-12 1983. (Revised by Wolf Lammen, 8-Jan-2018.) |
⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑) | ||
Theorem | excom13 1980 | Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑) | ||
Theorem | exrot3 1981 | Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑) | ||
Theorem | exrot4 1982 | Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑) | ||
Axiom | ax-12 1983 |
Axiom of Substitution. One of the 5 equality axioms of predicate
calculus. The final consequent ∀𝑥(𝑥 = 𝑦 → 𝜑) is a way of
expressing "𝑦 substituted for 𝑥 in wff
𝜑
" (cf. sb6 2321). 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-c15 33076 and was replaced with this shorter ax-12 1983 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2195. Conversely, this axiom is proved from ax-c15 33076 as theorem ax12 2196. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 33076) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. See ax12v 1984 and ax12v2 1985 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax12w 1958) but is used as an auxiliary axiom scheme to achieve scheme completeness. (Contributed by NM, 22-Jan-2007.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax12v 1984* |
This is essentially axiom ax-12 1983 weakened by additional restrictions on
variables. Besides axc11r 2136, this theorem should be the only one
referencing ax-12 1983 directly.
Both restrictions on variables have their own value. If for a moment we assume 𝑦 could be set to 𝑥, then, after elimination of the tautology 𝑥 = 𝑥, immediately we have 𝜑 → ∀𝑥𝜑 for all 𝜑 and 𝑥, that is ax-5 1793, a degenerate result. The second restriction is not necessary, but a simplification that makes the following interpretation easier to see. Since 𝜑 textually at most depends on 𝑥, we can look at it at some given 'fixed' 𝑦. This theorem now states that the truth value of 𝜑 will stay constant, as long as we 'vary 𝑥 around 𝑦' only such that 𝑥 = 𝑦 still holds. Or in other words, equality is the finest grained logical expression. If you cannot differ two sets by =, you won't find a whatever sophisticated expression that does. One might wonder how the described variation of 𝑥 is possible at all. Note that Metamath is a text processor that easily sees a difference between text chunks {𝑥 ∣ ¬ 𝑥 = 𝑥} and {𝑦 ∣ ¬ 𝑦 = 𝑦}. Our usual interpretation is to abstract from textual variations of the same set, but we are free to interpret Metamath's formalism differently, and in fact let 𝑥 run through all textual representations of sets. Had we allowed 𝜑 to depend also on 𝑦, this idea is both harder to see, and it is less clear that this extra freedom introduces effects not covered by other axioms. (Contributed by Wolf Lammen, 8-Aug-2020.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax12v2 1985* | It is possible to remove any restriction on 𝜑 in ax12v 1984. Same as Axiom C8 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1966 and ax-13 2137. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax12vOLD 1986* | Obsolete proof of ax12v2 1985 as of 24-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1966 and ax-13 2137. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof shortened by Wolf Lammen, 7-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax12vOLDOLD 1987* | Obsolete proof of ax12v 1984 as of 7-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1966 and ax-13 2137. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | 19.8a 1988 | If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 1845 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 1990. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) |
⊢ (𝜑 → ∃𝑥𝜑) | ||
Theorem | 19.8aOLD 1989 | Obsolete proof of 19.8a 1988. Obsolete as of 21-Dec-2020. Can be deleted as soon as the question of why "MM-PA> min exlimiiv" does not give 19.8a 1988 is answered. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 1990. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥𝜑) | ||
Theorem | sp 1990 |
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). This corresponds to the
axiom (T) of modal logic.
For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2245. This theorem shows that our obsolete axiom ax-c5 33070 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 auxiliary axiom scheme ax-12 1983. It is thought the best we can do using only Tarski's axioms is spw 1916. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | axc4 1991 |
Show that the original axiom ax-c4 33071 can be derived from ax-4 1713
(alim 1714), ax-10 1966 (hbn1 1967), sp 1990 and propositional calculus. See
ax4fromc4 33081 for the rederivation of ax-4 1713
from ax-c4 33071.
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.) |
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | axc7 1992 |
Show that the original axiom ax-c7 33072 can be derived from ax-10 1966
(hbn1 1967) , sp 1990 and propositional calculus. See ax10fromc7 33082 for the
rederivation of ax-10 1966 from ax-c7 33072.
Normally, axc7 1992 should be used rather than ax-c7 33072, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.) |
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | ||
Theorem | axc7e 1993 | Abbreviated version of axc7 1992 using the existential quantifier. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃𝑥∀𝑥𝜑 → 𝜑) | ||
Theorem | modal-b 1994 | The analogue in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.) |
⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑) | ||
Theorem | spi 1995 | Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.) |
⊢ ∀𝑥𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | sps 1996 | Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | 2sp 1997 | A double specialization (see sp 1990). Another double specialization, closer to PM*11.1, is 2stdpc4 2246. (Contributed by BJ, 15-Sep-2018.) |
⊢ (∀𝑥∀𝑦𝜑 → 𝜑) | ||
Theorem | spsd 1998 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | 19.2g 1999 | Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1842 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.) |
⊢ (∀𝑥𝜑 → ∃𝑦𝜑) | ||
Theorem | 19.21bi 2000 | Inference form of 19.21 2036 and also deduction form of sp 1990. (Contributed by NM, 26-May-1993.) |
⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
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