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Theorem List for Metamath Proof Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbf2 1901 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

Theoremsb6x 1902 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremnfs1f 1903 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremsbequ5 1904 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ6 1905 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)

Theoremsbt 1906 A substitution into a theorem remains true. (See chvar 1879 and chvarv 2062 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequsb1 1907 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremequsb2 1908 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremsbied 1909 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1911). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbiedv 1910* Conversion of implicit substitution to explicit substitution (deduction version of sbie 1911). (Contributed by NM, 7-Jan-2017.)

Theoremsbie 1911 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

1.5.14  Theorems using axiom ax-11

Theoremequs5a 1912 A property related to substitution that unlike equs5 1944 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs5e 1913 A property related to substitution that unlike equs5 1944 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremsb4a 1914 A version of sb4 1946 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs45f 1915 Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb6f 1916 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb5f 1917 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb4e 1918 One direction of a simplified definition of substitution that unlike sb4 1946 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2a 1919 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2e 1920 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb3 1921 If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremnfs1 1922 If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

1.6  Predicate calculus with distinct variables

1.6.1  Derive the axiom of distinct variables ax-16

Theorema4imv 1923* A version of a4im 1868 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)

Theoremaev 1924* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1927. (Contributed by NM, 8-Nov-2006.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremaev-o 1925* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1927. Version of aev 1924 using ax-10o 1836. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremax16 1926* Theorem showing that ax-16 1927 is redundant if ax-17 1628 is included in the axiom system. The important part of the proof is provided by aev 1924.

See ax16ALT 1996 for an alternate proof that does not require ax-10 1678 or ax-12o 1664.

This theorem should not be referenced in any proof. Instead, use ax-16 1927 below so that theorems needing ax-16 1927 can be more easily identified. (Contributed by NM, 8-Nov-2006.) (New usage is discouraged.)

Axiomax-16 1927* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1628 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 4173), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1628; see theorem ax16 1926. Alternately, ax-17 1628 becomes logically redundant in the presence of this axiom, but without ax-17 1628 we lose the more powerful metalogic that results from being able to express the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). We retain ax-16 1927 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1628, which might be easier to study for some theoretical purposes. (Contributed by NM, 5-Aug-1993.)

Theoremax17eq 1928* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1628 considered as a metatheorem. Do not use it for later proofs - use ax-17 1628 instead, to avoid reference to the redundant axiom ax-16 1927.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq2 1929* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremdveeq2-o 1930* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 1929 using ax-11o 1941. (Contributed by NM, 2-Jan-2002.)

Theoremdveeq2ALT 1931* Version of dveeq2 1929 using ax-16 1927 instead of ax-17 1628. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.)

Theoremnd5 1932* A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)

Theoremexlimdv 1933* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)

Theoremexlimdd 1934 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremexlimddv 1935* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Theoremax11v2 1936* Recovery of ax-11o 1941 from ax11v 1991. This proof uses ax-10 1678 and ax-11 1624. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)

Theoremax11v2-o 1937* Recovery of ax-11o 1941 from ax11v 1991 without using ax-11o 1941. The hypothesis is even weaker than ax11v 1991, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax-11o 1941. (Contributed by NM, 2-Feb-2007.)

Theoremax11a2 1938* Derive ax-11o 1941 from a hypothesis in the form of ax-11 1624. ax-10 1678 and ax-11 1624 are used by the proof, but not ax-10o 1836 or ax-11o 1941. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)

Theoremax11a2-o 1939* Derive ax-11o 1941 from a hypothesis in the form of ax-11 1624, without using ax-11 1624 or ax-11o 1941. The hypothesis is even weaker than ax-11 1624, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax-11 1624, if we also hvae ax-10o 1836 which this proof uses . As theorem ax11 1942 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 1678 instead of ax-10o 1836. (Contributed by NM, 2-Feb-2007.)

1.6.2  Derive the obsolete axiom of variable substitution ax-11o

Theoremax11o 1940 Derivation of set.mm's original ax-11o 1941 from ax-10 1678 and the shorter ax-11 1624 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1927 or ax-17 1628 (given all of the original and new versions of ax-4 1692 through ax-15 2106).

Another open problem is whether this theorem can be proved without relying on ax-12o 1664.

Theorem ax11 1942 shows the reverse derivation of ax-11 1624 from ax-11o 1941.

Normally, ax11o 1940 should be used rather than ax-11o 1941, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Axiomax-11o 1941 Axiom ax-11o 1941 ("o" for "old") was the original version of ax-11 1624, before it was discovered (in Jan. 2007) that the shorter ax-11 1624 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). 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. To understand this theorem more easily, think of " ..." as informally meaning "if and are distinct variables then..." The antecedent becomes false if the same variable is substituted for and , ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form a "distinctor."

This axiom is redundant, as shown by theorem ax11o 1940.

Normally, ax11o 1940 should be used rather than ax-11o 1941, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax11 1942 Rederivation of axiom ax-11 1624 from the orginal version, ax-11o 1941, and ax-10o 1836. The proof does not require ax-11 1624, ax-16 1927, or ax-17 1628. See theorem ax11o 1940 for the derivation of ax-11o 1941 from ax-11 1624.

An open problem is whether we can prove this using ax-10 1678 instead of ax-10o 1836.

This theorem should not be referenced in any proof. Instead, use ax-11 1624 above so that uses of ax-11 1624 can be more easily identified. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

1.6.3  Theorems without distinct variables that use axiom ax-11o

Theoremax11b 1943 A bidirectional version of ax-11o 1941. (Contributed by NM, 30-Jun-2006.)

Theoremequs5 1944 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Theoremsb3 1945 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4 1946 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4b 1947 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)

Theoremdfsb2 1948 An alternate definition of proper substitution that, like df-sb 1884, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)

Theoremdfsb3 1949 An alternate definition of proper substitution df-sb 1884 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)

Theoremhbsb2 1950 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremnfsb2 1951 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremsbequi 1952 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ 1953 An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremdrsb2 1954 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

Theoremsbn 1955 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsbi1 1956 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbi2 1957 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbim 1958 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsbor 1959 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)

Theoremsbrim 1960 Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsblim 1961 Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsban 1962 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsb3an 1963 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)

Theoremsbbi 1964 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsblbis 1965 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)

Theoremsbrbis 1966 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)

Theoremsbrbif 1967 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theorema4sbe 1968 A specialization theorem. (Contributed by NM, 5-Aug-1993.)

Theorema4sbim 1969 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorema4sbbi 1970 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsbbid 1971 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ8 1972 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)

Theoremnfsb4t 1973 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 1974). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremnfsb4 1974 A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremdvelimh 1975 Version of dvelim 2096 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)

Theoremdvelimf 1976 Version of dvelimv 2097 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremdvelimdf 1977 Deduction form of dvelimf 1976. This version may be useful if we want to avoid ax-17 1628 and use ax-16 1927 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbco 1978 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid2 1979 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbidm 1980 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsbco2 1981 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbco2d 1982 A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbco3 1983 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbcom 1984 A commutativity law for substitution. (Contributed by NM, 27-May-1997.)

Theoremsb5rf 1985 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb6rf 1986 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb8 1987 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb8e 1988 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb9i 1989 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)

Theoremsb9 1990 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)

1.6.4  Predicate calculus with distinct variables (cont.)

Theoremax11v 1991* This is a version of ax-11o 1941 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 1936 for the rederivation of ax-11o 1941 from this theorem. (Contributed by NM, 5-Aug-1993.)

Theoremsb56 1992* Two equivalent ways of expressing the proper substitution of for in , when and are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1884. (Contributed by NM, 14-Apr-2008.)

Theoremsb6 1993* Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.)

Theoremsb5 1994* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)

Theoremax16i 1995* Inference with ax-16 1927 as its conclusion, that doesn't require ax-10 1678, ax-11 1624, or ax-12o 1664 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)

Theoremax16ALT 1996* Version of ax16 1926 that doesn't require ax-10 1678 or ax-12o 1664 for its proof. (Contributed by NM, 17-May-2008.)

Theorema4v 1997* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)

Theorema4imev 1998* Distinct-variable version of a4ime 1869. (Contributed by NM, 5-Aug-1993.)

Theorema4eiv 1999* Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)

Theoremequvin 2000* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)

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