P. 20 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1901-2000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sb6 1901*	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.) (Revised by NM, 14-Apr-2008.)
|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
Theorem	sb5 1902*	Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
|- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
Theorem	sbnv 1903*	Version of sbn 1971 where x and y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
|- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
Theorem	sbanv 1904*	Version of sban 1974 where x and y are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
|- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
Theorem	sborv 1905*	Version of sbor 1973 where x and y are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
|- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
Theorem	sbi1v 1906*	Forward direction of sbimv 1908. (Contributed by Jim Kingdon, 25-Dec-2017.)
|- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
Theorem	sbi2v 1907*	Reverse direction of sbimv 1908. (Contributed by Jim Kingdon, 18-Jan-2018.)
|- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )
Theorem	sbimv 1908*	Intuitionistic proof of sbim 1972 where x and y are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)
|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
Theorem	sblimv 1909*	Version of sblim 1976 where x and y are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
|- ( ps -> A. x ps )   =>    |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )
Theorem	pm11.53 1910*	Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )
Theorem	exlimivv 1911*	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
|- ( ph -> ps )   =>    |- ( E. x E. y ph -> ps )
Theorem	exlimdvv 1912*	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x E. y ps -> ch ) )
Theorem	exlimddv 1913*	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
|- ( ph -> E. x ps )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> ch )
Theorem	19.27v 1914*	Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) )
Theorem	19.28v 1915*	Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
|- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) )
Theorem	19.36aiv 1916*	Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- E. x ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	19.41v 1917*	Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )
Theorem	19.41vv 1918*	Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> ( E. x E. y ph /\ ps ) )
Theorem	19.41vvv 1919*	Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
|- ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )
Theorem	19.41vvvv 1920*	Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
|- ( E. w E. x E. y E. z ( ph /\ ps ) <-> ( E. w E. x E. y E. z ph /\ ps ) )
Theorem	19.42v 1921*	Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
Theorem	spvv 1922*	Version of spv 1874 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	chvarvv 1923*	Version of chvarv 1956 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	exdistr 1924*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
Theorem	exdistrv 1925*	Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1917 and 19.42v 1921. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 1951. (Contributed by BJ, 30-Sep-2022.)
|- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )
Theorem	19.42vv 1926*	Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> ( ph /\ E. x E. y ps ) )
Theorem	19.42vvv 1927*	Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
|- ( E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. x E. y E. z ps ) )
Theorem	19.42vvvv 1928*	Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
|- ( E. w E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. w E. x E. y E. z ps ) )
Theorem	exdistr2 1929*	Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
|- ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( ph /\ E. y E. z ps ) )
Theorem	3exdistr 1930*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> E. x ( ph /\ E. y ( ps /\ E. z ch ) ) )
Theorem	4exdistr 1931*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )
Theorem	cbvalv 1932*	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbvexv 1933*	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	cbvalvw 1934*	Change bound variable. See cbvalv 1932 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1462. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbvexvw 1935*	Change bound variable. See cbvexv 1933 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1462. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	cbval2 1936*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
|- F/ z ph   &    |- F/ w ph   &    |- F/ x ps   &    |- F/ y ps   &    |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( A. x A. y ph <-> A. z A. w ps )
Theorem	cbvex2 1937*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- F/ z ph   &    |- F/ w ph   &    |- F/ x ps   &    |- F/ y ps   &    |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( E. x E. y ph <-> E. z E. w ps )
Theorem	cbval2v 1938*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( A. x A. y ph <-> A. z A. w ps )
Theorem	cbvex2v 1939*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( E. x E. y ph <-> E. z E. w ps )
Theorem	cbvald 1940*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2036. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
|- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( A. x ps <-> A. y ch ) )
Theorem	cbvexdh 1941*	Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 2036. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
|- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. y ps ) )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( E. x ps <-> E. y ch ) )
Theorem	cbvexd 1942*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2036. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
|- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( E. x ps <-> E. y ch ) )
Theorem	cbvaldva 1943*	Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. y ch ) )
Theorem	cbvexdva 1944*	Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. y ch ) )
Theorem	cbvaldvaw 1945*	Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 1943 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. y ch ) )
Theorem	cbvexdvaw 1946*	Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 1944 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. y ch ) )
Theorem	cbval2vw 1947*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) (Revised by GG, 10-Jan-2024.)
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( A. x A. y ph <-> A. z A. w ps )
Theorem	cbvex2vw 1948*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) (Revised by GG, 10-Jan-2024.)
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( E. x E. y ph <-> E. z E. w ps )
Theorem	cbvex4v 1949*	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 26-Jul-1995.)
|- ( ( x = v /\ y = u ) -> ( ph <-> ps ) )   &    |- ( ( z = f /\ w = g ) -> ( ps <-> ch ) )   =>    |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch )
Theorem	eean 1950	Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )
Theorem	eeanv 1951*	Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )
Theorem	eeeanv 1952*	Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> ( E. x ph /\ E. y ps /\ E. z ch ) )
Theorem	ee4anv 1953*	Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
|- ( E. x E. y E. z E. w ( ph /\ ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )
Theorem	ee8anv 1954*	Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
|- ( E. x E. y E. z E. w E. v E. u E. t E. s ( ph /\ ps ) <-> ( E. x E. y E. z E. w ph /\ E. v E. u E. t E. s ps ) )
Theorem	nexdv 1955*	Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> -. ps )   =>    |- ( ph -> -. E. x ps )
Theorem	chvarv 1956*	Implicit substitution of y for x into a theorem. (Contributed by NM, 20-Apr-1994.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
1.4.5  More substitution theorems
Theorem	hbs1 1957*	x is not free in [ y / x ] ph when x and y are distinct. (Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.) (New usage is discouraged.)
|- ( [ y / x ] ph -> A. x [ y / x ] ph )
Theorem	nfs1v 1958*	x is not free in [ y / x ] ph when x and y are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x [ y / x ] ph
Theorem	sbhb 1959*	Two ways of expressing " x is (effectively) not free in ph." (Contributed by NM, 29-May-2009.)
|- ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) )
Theorem	hbsbv 1960*	This is a version of hbsb 1968 with an extra distinct variable constraint, on z and x. (Contributed by Jim Kingdon, 25-Dec-2017.)
|- ( ph -> A. z ph )   =>    |- ( [ y / x ] ph -> A. z [ y / x ] ph )
Theorem	nfsbxy 1961*	Similar to hbsb 1968 but with an extra distinct variable constraint, on x and y. (Contributed by Jim Kingdon, 19-Mar-2018.)
|- F/ z ph   =>    |- F/ z [ y / x ] ph
Theorem	nfsbxyt 1962*	Closed form of nfsbxy 1961. (Contributed by Jim Kingdon, 9-May-2018.)
|- ( A. x F/ z ph -> F/ z [ y / x ] ph )
Theorem	sbco2vlem 1963*	This is a version of sbco2 1984 where z is distinct from x and from y. It is a lemma on the way to proving sbco2v 1967 which only requires that z and x be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) Remove one disjoint variable condition. (Revised by Jim Kingdon, 3-Feb-2018.)
|- ( ph -> A. z ph )   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	sbco2vh 1964*	This is a version of sbco2 1984 where z is distinct from x. (Contributed by Jim Kingdon, 12-Feb-2018.)
|- ( ph -> A. z ph )   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	nfsb 1965*	If z is not free in ph, it is not free in [ y / x ] ph when y and z are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
|- F/ z ph   =>    |- F/ z [ y / x ] ph
Theorem	nfsbv 1966*	If z is not free in ph, it is not free in [ y / x ] ph when z is distinct from x and y. Version of nfsb 1965 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on x , y. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
|- F/ z ph   =>    |- F/ z [ y / x ] ph
Theorem	sbco2v 1967*	Version of sbco2 1984 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)
|- F/ z ph   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	hbsb 1968*	If z is not free in ph, it is not free in [ y / x ] ph when y and z are distinct. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
|- ( ph -> A. z ph )   =>    |- ( [ y / x ] ph -> A. z [ y / x ] ph )
Theorem	equsb3lem 1969*	Lemma for equsb3 1970. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x = z <-> y = z )
Theorem	equsb3 1970*	Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
|- ( [ y / x ] x = z <-> y = z )
Theorem	sbn 1971	Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
|- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
Theorem	sbim 1972	Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
Theorem	sbor 1973	Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
|- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
Theorem	sban 1974	Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
|- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
Theorem	sbrim 1975	Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )
Theorem	sblim 1976	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.)
|- F/ x ps   =>    |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )
Theorem	sb3an 1977	Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
|- ( [ y / x ] ( ph /\ ps /\ ch ) <-> ( [ y / x ] ph /\ [ y / x ] ps /\ [ y / x ] ch ) )
Theorem	sbbi 1978	Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ( ph <-> ps ) <-> ( [ y / x ] ph <-> [ y / x ] ps ) )
Theorem	sblbis 1979	Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
|- ( [ y / x ] ph <-> ps )   =>    |- ( [ y / x ] ( ch <-> ph ) <-> ( [ y / x ] ch <-> ps ) )
Theorem	sbrbis 1980	Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
|- ( [ y / x ] ph <-> ps )   =>    |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> [ y / x ] ch ) )
Theorem	sbrbif 1981	Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
|- ( ch -> A. x ch )   &    |- ( [ y / x ] ph <-> ps )   =>    |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> ch ) )
Theorem	sbco2yz 1982*	This is a version of sbco2 1984 where z is distinct from y. It is a lemma on the way to proving sbco2 1984 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
|- F/ z ph   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	sbco2h 1983	A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
|- ( ph -> A. z ph )   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	sbco2 1984	A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- F/ z ph   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	sbco2d 1985	A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> A. z ph )   &    |- ( ph -> ( ps -> A. z ps ) )   =>    |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )
Theorem	sbco2vd 1986*	Version of sbco2d 1985 with a distinct variable constraint between x and z. (Contributed by Jim Kingdon, 19-Feb-2018.)
|- ( ph -> A. x ph )   &    |- ( ph -> A. z ph )   &    |- ( ph -> ( ps -> A. z ps ) )   =>    |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )
Theorem	sbco 1987	A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
Theorem	sbco3v 1988*	Version of sbco3 1993 with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)
|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
Theorem	sbcocom 1989	Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.)
|- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )
Theorem	sbcomv 1990*	Version of sbcom 1994 with a distinct variable constraint between x and z. (Contributed by Jim Kingdon, 28-Feb-2018.)
|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )
Theorem	sbcomxyyz 1991*	Version of sbcom 1994 with distinct variable constraints between x and y, and y and z. (Contributed by Jim Kingdon, 21-Mar-2018.)
|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )
Theorem	sbco3xzyz 1992*	Version of sbco3 1993 with distinct variable constraints between x and z, and y and z. Lemma for proving sbco3 1993. (Contributed by Jim Kingdon, 22-Mar-2018.)
|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
Theorem	sbco3 1993	A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
Theorem	sbcom 1994	A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )
Theorem	nfsbt 1995*	Closed form of nfsb 1965. (Contributed by Jim Kingdon, 9-May-2018.)
|- ( A. x F/ z ph -> F/ z [ y / x ] ph )
Theorem	nfsbd 1996*	Deduction version of nfsb 1965. (Contributed by NM, 15-Feb-2013.)
|- F/ x ph   &    |- ( ph -> F/ z ps )   =>    |- ( ph -> F/ z [ y / x ] ps )
Theorem	sb9v 1997*	Like sb9 1998 but with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 28-Feb-2018.)
|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )
Theorem	sb9 1998	Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )
Theorem	sb9i 1999	Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
|- ( A. x [ x / y ] ph -> A. y [ y / x ] ph )
Theorem	sbnf2 2000*	Two ways of expressing " x is (effectively) not free in ph." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- ( F/ x ph <-> A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) )