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	sb8eh 1901	Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
|- ( ph -> A. y ph )   =>    |- ( E. x ph <-> E. y [ y / x ] ph )
Theorem	sb8 1902	Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
|- F/ y ph   =>    |- ( A. x ph <-> A. y [ y / x ] ph )
Theorem	sb8e 1903	Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
|- F/ y ph   =>    |- ( E. x ph <-> E. y [ y / x ] ph )
1.4.4  Predicate calculus with distinct variables (cont.)
Theorem	ax16i 1904*	Inference with ax-16 1860 as its conclusion, that does not require ax-10 1551, ax-11 1552, or ax12 1558 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)
|- ( x = z -> ( ph <-> ps ) )   &    |- ( ps -> A. x ps )   =>    |- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	ax16ALT 1905*	Version of ax16 1859 that does not require ax-10 1551 or ax12 1558 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	spv 1906*	Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spimev 1907*	Distinct-variable version of spime 1787. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	speiv 1908*	Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ps   =>    |- E. x ph
Theorem	equvin 1909*	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
|- ( x = y <-> E. z ( x = z /\ z = y ) )
Theorem	a16g 1910*	A generalization of Axiom ax-16 1860. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( A. x x = y -> ( ph -> A. z ph ) )
Theorem	a16gb 1911*	A generalization of Axiom ax-16 1860. (Contributed by NM, 5-Aug-1993.)
|- ( A. x x = y -> ( ph <-> A. z ph ) )
Theorem	a16nf 1912*	If there is only one element in the universe, then everything satisfies F/. (Contributed by Mario Carneiro, 7-Oct-2016.)
|- ( A. x x = y -> F/ z ph )
Theorem	2albidv 1913*	Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x A. y ps <-> A. x A. y ch ) )
Theorem	2exbidv 1914*	Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )
Theorem	3exbidv 1915*	Formula-building rule for 3 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) )
Theorem	4exbidv 1916*	Formula-building rule for 4 existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y E. z E. w ps <-> E. x E. y E. z E. w ch ) )
Theorem	19.9v 1917*	Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
|- ( E. x ph <-> ph )
Theorem	exlimdd 1918	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- F/ x ph   &    |- F/ x ch   &    |- ( ph -> E. x ps )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> ch )
Theorem	19.21v 1919*	Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as ( ph -> A. x ph ) in 19.21 1629 via the use of distinct variable conditions combined with ax-17 1572. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g., euf 2082 derived from df-eu 2080. The "f" stands for "not free in" which is less restrictive than "does not occur in". (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
Theorem	alrimiv 1920*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	alrimivv 1921*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
|- ( ph -> ps )   =>    |- ( ph -> A. x A. y ps )
Theorem	alrimdv 1922*	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	nfdv 1923*	Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	2ax17 1924*	Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
|- ( ph -> A. x A. y ph )
Theorem	alimdv 1925*	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	eximdv 1926*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> E. x ch ) )
Theorem	2alimdv 1927*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x A. y ps -> A. x A. y ch ) )
Theorem	2eximdv 1928*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x E. y ps -> E. x E. y ch ) )
Theorem	19.23v 1929*	Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
|- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	19.23vv 1930*	Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
|- ( A. x A. y ( ph -> ps ) <-> ( E. x E. y ph -> ps ) )
Theorem	sbbidv 1931*	Deduction substituting both sides of a biconditional, with ph and x disjoint. See also sbbid 1892. (Contributed by Wolf Lammen, 6-May-2023.) (Proof shortened by Steven Nguyen, 6-Jul-2023.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ t / x ] ps <-> [ t / x ] ch ) )
Theorem	sb56 1932*	Two equivalent ways of expressing the proper substitution of y for x in ph, when x and y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1809. (Contributed by NM, 14-Apr-2008.)
|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
Theorem	sb6 1933*	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 1934*	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 1935*	Version of sbn 2003 where x and y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
|- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
Theorem	sbanv 1936*	Version of sban 2006 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 1937*	Version of sbor 2005 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 1938*	Forward direction of sbimv 1940. (Contributed by Jim Kingdon, 25-Dec-2017.)
|- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
Theorem	sbi2v 1939*	Reverse direction of sbimv 1940. (Contributed by Jim Kingdon, 18-Jan-2018.)
|- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )
Theorem	sbimv 1940*	Intuitionistic proof of sbim 2004 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 1941*	Version of sblim 2008 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 1942*	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 1943*	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 1944*	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 1945*	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 1946*	Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) )
Theorem	19.28v 1947*	Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
|- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) )
Theorem	19.36aiv 1948*	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 1949*	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 1950*	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 1951*	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 1952*	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 1953*	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 1954*	Version of spv 1906 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	chvarvv 1955*	Version of chvarv 1988 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	exdistr 1956*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
Theorem	exdistrv 1957*	Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1949 and 19.42v 1953. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 1983. (Contributed by BJ, 30-Sep-2022.)
|- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )
Theorem	19.42vv 1958*	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 1959*	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 1960*	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 1961*	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 1962*	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 1963*	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 1964*	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 1965*	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 1966*	Change bound variable. See cbvalv 1964 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1494. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbvexvw 1967*	Change bound variable. See cbvexv 1965 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1494. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	cbval2 1968*	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 1969*	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 1970*	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 1971*	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 1972*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2068. (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 1973*	Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 2068. (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 1974*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2068. (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 1975*	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 1976*	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 1977*	Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 1975 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 1978*	Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 1976 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 1979*	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 1980*	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 1981*	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 1982	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 1983*	Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )
Theorem	eeeanv 1984*	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 1985*	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 1986*	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 1987*	Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> -. ps )   =>    |- ( ph -> -. E. x ps )
Theorem	chvarv 1988*	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 1989*	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 1990*	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 1991*	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 1992*	This is a version of hbsb 2000 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 1993*	Similar to hbsb 2000 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 1994*	Closed form of nfsbxy 1993. (Contributed by Jim Kingdon, 9-May-2018.)
|- ( A. x F/ z ph -> F/ z [ y / x ] ph )
Theorem	sbco2vlem 1995*	This is a version of sbco2 2016 where z is distinct from x and from y. It is a lemma on the way to proving sbco2v 1999 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 1996*	This is a version of sbco2 2016 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 1997*	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 1998*	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 1997 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 1999*	Version of sbco2 2016 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)
|- F/ z ph   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
Theorem	hbsb 2000*	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 )