mmtheorems20 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1901-2000 - Page 20 of 175

Type	Label	Description
Statement
Theorem	eqeqan12d 1901	A useful inference for substituting definitions into an equality. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ph /\ ps) -> (A = C <-> B = D))
Theorem	eqeqan12dOLD 1902	A useful inference for substituting definitions into an equality.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ph /\ ps) -> (A = C <-> B = D))
Theorem	eqeqan12rd 1903	A useful inference for substituting definitions into an equality.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ps /\ ph) -> (A = C <-> B = D))
Theorem	eqtr 1904	Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13.
|- ((A = B /\ B = C) -> A = C)
Theorem	eqtr2 1905	A transitive law for class equality. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- ((A = B /\ A = C) -> B = C)
Theorem	eqtr2OLD 1906	A transitive law for class equality.
|- ((A = B /\ A = C) -> B = C)
Theorem	eqtr3 1907	A transitive law for class equality.
|- ((A = C /\ B = C) -> A = B)
Theorem	eqtri 1908	An equality transitivity inference.
|- A = B   &   |- B = C   =>   |- A = C
Theorem	eqtr2i 1909	An equality transitivity inference.
|- A = B   &   |- B = C   =>   |- C = A
Theorem	eqtr3i 1910	An equality transitivity inference.
|- A = B   &   |- A = C   =>   |- B = C
Theorem	eqtr4i 1911	An equality transitivity inference.
|- A = B   &   |- C = B   =>   |- A = C
Theorem	3eqtri 1912	An inference from three chained equalities.
|- A = B   &   |- B = C   &   |- C = D   =>   |- A = D
Theorem	3eqtrri 1913	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- B = C   &   |- C = D   =>   |- D = A
Theorem	3eqtrriOLD 1914	An inference from three chained equalities.
|- A = B   &   |- B = C   &   |- C = D   =>   |- D = A
Theorem	3eqtr2i 1915	An inference from three chained equalities.
|- A = B   &   |- C = B   &   |- C = D   =>   |- A = D
Theorem	3eqtr2ri 1916	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- C = B   &   |- C = D   =>   |- D = A
Theorem	3eqtr2riOLD 1917	An inference from three chained equalities.
|- A = B   &   |- C = B   &   |- C = D   =>   |- D = A
Theorem	3eqtr3i 1918	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- A = C   &   |- B = D   =>   |- C = D
Theorem	3eqtr3iOLD 1919	An inference from three chained equalities.
|- A = B   &   |- A = C   &   |- B = D   =>   |- C = D
Theorem	3eqtr3ri 1920	An inference from three chained equalities.
|- A = B   &   |- A = C   &   |- B = D   =>   |- D = C
Theorem	3eqtr4i 1921	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- C = A   &   |- D = B   =>   |- C = D
Theorem	3eqtr4iOLD 1922	An inference from three chained equalities.
|- A = B   &   |- C = A   &   |- D = B   =>   |- C = D
Theorem	3eqtr4ri 1923	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- C = A   &   |- D = B   =>   |- D = C
Theorem	3eqtr4riOLD 1924	An inference from three chained equalities.
|- A = B   &   |- C = A   &   |- D = B   =>   |- D = C
Theorem	eqtrd 1925	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ph -> B = C)   =>   |- (ph -> A = C)
Theorem	eqtr2d 1926	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ph -> B = C)   =>   |- (ph -> C = A)
Theorem	eqtr3d 1927	An equality transitivity equality deduction.
|- (ph -> A = B)   &   |- (ph -> A = C)   =>   |- (ph -> B = C)
Theorem	eqtr4d 1928	An equality transitivity equality deduction.
|- (ph -> A = B)   &   |- (ph -> C = B)   =>   |- (ph -> A = C)
Theorem	3eqtrd 1929	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> B = C)   &   |- (ph -> C = D)   =>   |- (ph -> A = D)
Theorem	3eqtrrd 1930	A deduction from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ph -> B = C)   &   |- (ph -> C = D)   =>   |- (ph -> D = A)
Theorem	3eqtrrdOLD 1931	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> B = C)   &   |- (ph -> C = D)   =>   |- (ph -> D = A)
Theorem	3eqtr2d 1932	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = B)   &   |- (ph -> C = D)   =>   |- (ph -> A = D)
Theorem	3eqtr2rd 1933	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = B)   &   |- (ph -> C = D)   =>   |- (ph -> D = A)
Theorem	3eqtr3d 1934	A deduction from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> C = D)
Theorem	3eqtr3dOLD 1935	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> C = D)
Theorem	3eqtr3rd 1936	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> D = C)
Theorem	3eqtr4d 1937	A deduction from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C = D)
Theorem	3eqtr4dOLD 1938	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C = D)
Theorem	3eqtr4rd 1939	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> D = C)
Theorem	syl5eq 1940	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = A   =>   |- (ph -> C = B)
Theorem	syl5req 1941	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = A   =>   |- (ph -> B = C)
Theorem	syl5eqr 1942	An equality transitivity deduction.
|- (ph -> A = B)   &   |- A = C   =>   |- (ph -> C = B)
Theorem	syl5reqr 1943	An equality transitivity deduction.
|- (ph -> A = B)   &   |- A = C   =>   |- (ph -> B = C)
Theorem	syl6eq 1944	An equality transitivity deduction.
|- (ph -> A = B)   &   |- B = C   =>   |- (ph -> A = C)
Theorem	syl6req 1945	An equality transitivity deduction.
|- (ph -> A = B)   &   |- B = C   =>   |- (ph -> C = A)
Theorem	syl6eqr 1946	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = B   =>   |- (ph -> A = C)
Theorem	syl6reqr 1947	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = B   =>   |- (ph -> C = A)
Theorem	sylan9eq 1948	An equality transitivity deduction. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9eqOLD 1949	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9req 1950	An equality transitivity deduction.
|- (ph -> B = A)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9eqr 1951	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ps /\ ph) -> A = C)
Theorem	3eqtr3g 1952	A chained equality inference, useful for converting from definitions.
|- (ph -> A = B)   &   |- A = C   &   |- B = D   =>   |- (ph -> C = D)
Theorem	3eqtr4g 1953	A chained equality inference, useful for converting to definitions.
|- (ph -> A = B)   &   |- C = A   &   |- D = B   =>   |- (ph -> C = D)
Theorem	3eqtr4a 1954	A chained equality inference, useful for converting to definitions. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C = D)
Theorem	3eqtr4aOLD 1955	A chained equality inference, useful for converting to definitions.
|- A = B   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C = D)
Theorem	eq2tri 1956	A compound transitive inference for class equality.
|- (A = C -> D = F)   &   |- (B = D -> C = G)   =>   |- ((A = C /\ B = F) <-> (B = D /\ A = G))
Theorem	eleq1 1957	Equality implies equivalence of membership.
|- (A = B -> (A e. C <-> B e. C))
Theorem	eleq2 1958	Equality implies equivalence of membership.
|- (A = B -> (C e. A <-> C e. B))
Theorem	eleq12 1959	Equality implies equivalence of membership.
|- ((A = B /\ C = D) -> (A e. C <-> B e. D))
Theorem	eleq1i 1960	Inference from equality to equivalence of membership.
|- A = B   =>   |- (A e. C <-> B e. C)
Theorem	eleq2i 1961	Inference from equality to equivalence of membership.
|- A = B   =>   |- (C e. A <-> C e. B)
Theorem	eleq12i 1962	Inference from equality to equivalence of membership.
|- A = B   &   |- C = D   =>   |- (A e. C <-> B e. D)
Theorem	eleq1d 1963	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   =>   |- (ph -> (A e. C <-> B e. C))
Theorem	eleq2d 1964	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   =>   |- (ph -> (C e. A <-> C e. B))
Theorem	eleq12d 1965	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A e. C <-> B e. D))
Theorem	eleq1a 1966	A transitive-type law relating membership and equality.
|- (A e. B -> (C = A -> C e. B))
Theorem	eqeltri 1967	Substitution of equal classes into membership relation.
|- A = B   &   |- B e. C   =>   |- A e. C
Theorem	eqeltrri 1968	Substitution of equal classes into membership relation.
|- A = B   &   |- A e. C   =>   |- B e. C
Theorem	eleqtri 1969	Substitution of equal classes into membership relation.
|- A e. B   &   |- B = C   =>   |- A e. C
Theorem	eleqtrri 1970	Substitution of equal classes into membership relation.
|- A e. B   &   |- C = B   =>   |- A e. C
Theorem	eqeltrd 1971	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
|- (ph -> A = B)   &   |- (ph -> B e. C)   =>   |- (ph -> A e. C)
Theorem	eqeltrrd 1972	Deduction that substitutes equal classes into membership.
|- (ph -> A = B)   &   |- (ph -> A e. C)   =>   |- (ph -> B e. C)
Theorem	eleqtrd 1973	Deduction that substitutes equal classes into membership.
|- (ph -> A e. B)   &   |- (ph -> B = C)   =>   |- (ph -> A e. C)
Theorem	eleqtrrd 1974	Deduction that substitutes equal classes into membership.
|- (ph -> A e. B)   &   |- (ph -> C = B)   =>   |- (ph -> A e. C)
Theorem	syl5eqel 1975	A membership and equality inference.
|- (ph -> A e. B)   &   |- C = A   =>   |- (ph -> C e. B)
Theorem	syl5eqelr 1976	A membership and equality inference.
|- (ph -> A e. B)   &   |- A = C   =>   |- (ph -> C e. B)
Theorem	syl5eleq 1977	A membership and equality inference.
|- (ph -> A = B)   &   |- C e. A   =>   |- (ph -> C e. B)
Theorem	syl5eleqr 1978	A membership and equality inference.
|- (ph -> B = A)   &   |- C e. A   =>   |- (ph -> C e. B)
Theorem	syl6eqel 1979	A membership and equality inference.
|- (ph -> A = B)   &   |- B e. C   =>   |- (ph -> A e. C)
Theorem	syl6eqelr 1980	A membership and equality inference.
|- (ph -> B = A)   &   |- B e. C   =>   |- (ph -> A e. C)
Theorem	syl6eleq 1981	A membership and equality inference.
|- (ph -> A e. B)   &   |- B = C   =>   |- (ph -> A e. C)
Theorem	syl6eleqr 1982	A membership and equality inference.
|- (ph -> A e. B)   &   |- C = B   =>   |- (ph -> A e. C)
Theorem	eleq2s 1983	Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- (A e. B -> ph)   &   |- C = B   =>   |- (A e. C -> ph)
Theorem	cleqf 1984	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (A = B <-> A.x(x e. A <-> x e. B))
Theorem	nelneq 1985	A way of showing two classes are not equal.
|- ((A e. C /\ -. B e. C) -> -. A = B)
Theorem	nelneq2 1986	A way of showing two classes are not equal.
|- ((A e. B /\ -. A e. C) -> -. B = C)
Theorem	eqsb3lem 1987	Lemma for eqsb3 1989. (The proof was shortened by Andrew Salmon, 14-Jun-2011.)
Theorem	eqsb3lemOLD 1988	Lemma for eqsb3 1989.
Theorem	eqsb3 1989	Substitution applied to an atomic wff (class version of equsb3 1717). (Contributed by Rodolfo Medina, 28-Apr-2010.)
|- ([x / y]y = A <-> x = A)
Theorem	clelsb3 1990	Substitution applied to an atomic wff (class version of elsb3 1718). (Contributed by Rodolfo Medina, 28-Apr-2010.) (The proof was shortened by Andrew Salmon, 14-Jun-2011.)
|- ([x / y]y e. A <-> x e. A)
Theorem	clelsb3OLD 1991	Substitution applied to an atomic wff (class version of elsb3 1718). (Contributed by Rodolfo Medina, 28-Apr-2010.)
|- ([x / y]y e. A <-> x e. A)
Theorem	hbxfr 1992	A utility lemma to transfer a bound-variable hypothesis builder into a definition.
|- A = B   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. A -> A.x y e. A)
Theorem	hblem 1993	Lemma for hbeq 1995 and hbel 1996. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
Theorem	hblemOLD 1994	Lemma for hbeq 1995 and hbel 1996.
Theorem	hbeq 1995	If x is effectively bound in A and B, it is effectively bound in A = B. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
|- (y e. A -> A.x y e. A)   &   |- (z e. B -> A.x z e. B)   =>   |- (A = B -> A.x A = B)
Theorem	hbel 1996	If x is effectively bound in A and B, it is effectively bound in A e. B. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
|- (y e. A -> A.x y e. A)   &   |- (z e. B -> A.x z e. B)   =>   |- (A e. B -> A.x A e. B)
Theorem	hbeleq 1997	If x is effectively bound in y e. A, then it is effectively bound in y = A. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
|- (y e. A -> A.x y e. A)   =>   |- (y = A -> A.x y = A)
Theorem	hbeleqOLD 1998	If x is effectively bound in y e. A, then it is effectively bound in y = A.
|- (y e. A -> A.x y e. A)   =>   |- (y = A -> A.x y = A)
Theorem	abeq2 1999	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that eq2ab 2004 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable ph (that has a free variable x) to a theorem with a class variable A, we substitute x e. A for ph throughout and simplify, where A is a new class variable not already in the wff. An example is the conversion of zfauscl 3440 to inex1 3452 (look at the instance of zfauscl 3440 that occurs in the proof of inex1 3452). Conversely, to convert a theorem with a class variable A to one with ph, we substitute {x | ph} for A throughout and simplify, where x and ph are new set and wff variables not already in the wff. An example is cp 5852, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 5851.
|- (A = {x | ph} <-> A.x(x e. A <-> ph))
Theorem	abeq1 2000	Equality of a class variable and a class abstraction.
|- ({x | ph} = A <-> A.x(ph <-> x e. A))