Type | Label | Description |
Statement |
|
Theorem | 19.36aiv 1901* |
Inference from Theorem 19.36 of [Margaris] p.
90. (Contributed by NM,
5-Aug-1993.)
|
β’ βπ₯(π β π) β β’ (βπ₯π β π) |
|
Theorem | 19.41v 1902* |
Special case of Theorem 19.41 of [Margaris] p.
90. (Contributed by NM,
5-Aug-1993.)
|
β’ (βπ₯(π β§ π) β (βπ₯π β§ π)) |
|
Theorem | 19.41vv 1903* |
Theorem 19.41 of [Margaris] p. 90 with 2
quantifiers. (Contributed by
NM, 30-Apr-1995.)
|
β’ (βπ₯βπ¦(π β§ π) β (βπ₯βπ¦π β§ π)) |
|
Theorem | 19.41vvv 1904* |
Theorem 19.41 of [Margaris] p. 90 with 3
quantifiers. (Contributed by
NM, 30-Apr-1995.)
|
β’ (βπ₯βπ¦βπ§(π β§ π) β (βπ₯βπ¦βπ§π β§ π)) |
|
Theorem | 19.41vvvv 1905* |
Theorem 19.41 of [Margaris] p. 90 with 4
quantifiers. (Contributed by
FL, 14-Jul-2007.)
|
β’ (βπ€βπ₯βπ¦βπ§(π β§ π) β (βπ€βπ₯βπ¦βπ§π β§ π)) |
|
Theorem | 19.42v 1906* |
Special case of Theorem 19.42 of [Margaris] p.
90. (Contributed by NM,
5-Aug-1993.)
|
β’ (βπ₯(π β§ π) β (π β§ βπ₯π)) |
|
Theorem | spvv 1907* |
Version of spv 1860 with a disjoint variable condition.
(Contributed by
BJ, 31-May-2019.)
|
β’ (π₯ = π¦ β (π β π)) β β’ (βπ₯π β π) |
|
Theorem | chvarvv 1908* |
Version of chvarv 1937 with a disjoint variable condition.
(Contributed by
BJ, 31-May-2019.)
|
β’ (π₯ = π¦ β (π β π)) & β’ π β β’ π |
|
Theorem | exdistr 1909* |
Distribution of existential quantifiers. (Contributed by NM,
9-Mar-1995.)
|
β’ (βπ₯βπ¦(π β§ π) β βπ₯(π β§ βπ¦π)) |
|
Theorem | exdistrv 1910* |
Distribute a pair of existential quantifiers (over disjoint variables)
over a conjunction. Combination of 19.41v 1902 and 19.42v 1906. For a
version with fewer disjoint variable conditions but requiring more
axioms, see eeanv 1932. (Contributed by BJ, 30-Sep-2022.)
|
β’ (βπ₯βπ¦(π β§ π) β (βπ₯π β§ βπ¦π)) |
|
Theorem | 19.42vv 1911* |
Theorem 19.42 of [Margaris] p. 90 with 2
quantifiers. (Contributed by
NM, 16-Mar-1995.)
|
β’ (βπ₯βπ¦(π β§ π) β (π β§ βπ₯βπ¦π)) |
|
Theorem | 19.42vvv 1912* |
Theorem 19.42 of [Margaris] p. 90 with 3
quantifiers. (Contributed by
NM, 21-Sep-2011.)
|
β’ (βπ₯βπ¦βπ§(π β§ π) β (π β§ βπ₯βπ¦βπ§π)) |
|
Theorem | 19.42vvvv 1913* |
Theorem 19.42 of [Margaris] p. 90 with 4
quantifiers. (Contributed by
Jim Kingdon, 23-Nov-2019.)
|
β’ (βπ€βπ₯βπ¦βπ§(π β§ π) β (π β§ βπ€βπ₯βπ¦βπ§π)) |
|
Theorem | exdistr2 1914* |
Distribution of existential quantifiers. (Contributed by NM,
17-Mar-1995.)
|
β’ (βπ₯βπ¦βπ§(π β§ π) β βπ₯(π β§ βπ¦βπ§π)) |
|
Theorem | 3exdistr 1915* |
Distribution of existential quantifiers. (Contributed by NM,
9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|
β’ (βπ₯βπ¦βπ§(π β§ π β§ π) β βπ₯(π β§ βπ¦(π β§ βπ§π))) |
|
Theorem | 4exdistr 1916* |
Distribution of existential quantifiers. (Contributed by NM,
9-Mar-1995.)
|
β’ (βπ₯βπ¦βπ§βπ€((π β§ π) β§ (π β§ π)) β βπ₯(π β§ βπ¦(π β§ βπ§(π β§ βπ€π)))) |
|
Theorem | cbvalv 1917* |
Rule used to change bound variables, using implicit substitition.
(Contributed by NM, 5-Aug-1993.)
|
β’ (π₯ = π¦ β (π β π)) β β’ (βπ₯π β βπ¦π) |
|
Theorem | cbvexv 1918* |
Rule used to change bound variables, using implicit substitition.
(Contributed by NM, 5-Aug-1993.)
|
β’ (π₯ = π¦ β (π β π)) β β’ (βπ₯π β βπ¦π) |
|
Theorem | cbvalvw 1919* |
Change bound variable. See cbvalv 1917 for a version with fewer disjoint
variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1448.
(Revised by Gino Giotto, 25-Aug-2024.)
|
β’ (π₯ = π¦ β (π β π)) β β’ (βπ₯π β βπ¦π) |
|
Theorem | cbvexvw 1920* |
Change bound variable. See cbvexv 1918 for a version with fewer disjoint
variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1448.
(Revised by Gino Giotto, 25-Aug-2024.)
|
β’ (π₯ = π¦ β (π β π)) β β’ (βπ₯π β βπ¦π) |
|
Theorem | cbval2 1921* |
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.)
|
β’ β²π§π
& β’ β²π€π
& β’ β²π₯π
& β’ β²π¦π
& β’ ((π₯ = π§ β§ π¦ = π€) β (π β π)) β β’ (βπ₯βπ¦π β βπ§βπ€π) |
|
Theorem | cbvex2 1922* |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro,
6-Oct-2016.)
|
β’ β²π§π
& β’ β²π€π
& β’ β²π₯π
& β’ β²π¦π
& β’ ((π₯ = π§ β§ π¦ = π€) β (π β π)) β β’ (βπ₯βπ¦π β βπ§βπ€π) |
|
Theorem | cbval2v 1923* |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 4-Feb-2005.)
|
β’ ((π₯ = π§ β§ π¦ = π€) β (π β π)) β β’ (βπ₯βπ¦π β βπ§βπ€π) |
|
Theorem | cbvex2v 1924* |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 26-Jul-1995.)
|
β’ ((π₯ = π§ β§ π¦ = π€) β (π β π)) β β’ (βπ₯βπ¦π β βπ§βπ€π) |
|
Theorem | cbvald 1925* |
Deduction used to change bound variables, using implicit substitution,
particularly useful in conjunction with dvelim 2017. (Contributed by NM,
2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf
Lammen, 13-May-2018.)
|
β’ β²π¦π
& β’ (π β β²π¦π)
& β’ (π β (π₯ = π¦ β (π β π))) β β’ (π β (βπ₯π β βπ¦π)) |
|
Theorem | cbvexdh 1926* |
Deduction used to change bound variables, using implicit substitition,
particularly useful in conjunction with dvelim 2017. (Contributed by NM,
2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
|
β’ (π β βπ¦π)
& β’ (π β (π β βπ¦π)) & β’ (π β (π₯ = π¦ β (π β π))) β β’ (π β (βπ₯π β βπ¦π)) |
|
Theorem | cbvexd 1927* |
Deduction used to change bound variables, using implicit substitution,
particularly useful in conjunction with dvelim 2017. (Contributed by NM,
2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten
by Jim Kingdon, 10-Jun-2018.)
|
β’ β²π¦π
& β’ (π β β²π¦π)
& β’ (π β (π₯ = π¦ β (π β π))) β β’ (π β (βπ₯π β βπ¦π)) |
|
Theorem | cbvaldva 1928* |
Rule used to change the bound variable in a universal quantifier with
implicit substitution. Deduction form. (Contributed by David Moews,
1-May-2017.)
|
β’ ((π β§ π₯ = π¦) β (π β π)) β β’ (π β (βπ₯π β βπ¦π)) |
|
Theorem | cbvexdva 1929* |
Rule used to change the bound variable in an existential quantifier with
implicit substitution. Deduction form. (Contributed by David Moews,
1-May-2017.)
|
β’ ((π β§ π₯ = π¦) β (π β π)) β β’ (π β (βπ₯π β βπ¦π)) |
|
Theorem | cbvex4v 1930* |
Rule used to change bound variables, using implicit substitition.
(Contributed by NM, 26-Jul-1995.)
|
β’ ((π₯ = π£ β§ π¦ = π’) β (π β π)) & β’ ((π§ = π β§ π€ = π) β (π β π)) β β’ (βπ₯βπ¦βπ§βπ€π β βπ£βπ’βπβππ) |
|
Theorem | eean 1931 |
Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
(Revised by Mario Carneiro, 6-Oct-2016.)
|
β’ β²π¦π
& β’ β²π₯π β β’ (βπ₯βπ¦(π β§ π) β (βπ₯π β§ βπ¦π)) |
|
Theorem | eeanv 1932* |
Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
|
β’ (βπ₯βπ¦(π β§ π) β (βπ₯π β§ βπ¦π)) |
|
Theorem | eeeanv 1933* |
Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
(Proof shortened by Andrew Salmon, 25-May-2011.)
|
β’ (βπ₯βπ¦βπ§(π β§ π β§ π) β (βπ₯π β§ βπ¦π β§ βπ§π)) |
|
Theorem | ee4anv 1934* |
Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
|
β’ (βπ₯βπ¦βπ§βπ€(π β§ π) β (βπ₯βπ¦π β§ βπ§βπ€π)) |
|
Theorem | ee8anv 1935* |
Rearrange existential quantifiers. (Contributed by Jim Kingdon,
23-Nov-2019.)
|
β’ (βπ₯βπ¦βπ§βπ€βπ£βπ’βπ‘βπ (π β§ π) β (βπ₯βπ¦βπ§βπ€π β§ βπ£βπ’βπ‘βπ π)) |
|
Theorem | nexdv 1936* |
Deduction for generalization rule for negated wff. (Contributed by NM,
5-Aug-1993.)
|
β’ (π β Β¬ π) β β’ (π β Β¬ βπ₯π) |
|
Theorem | chvarv 1937* |
Implicit substitution of π¦ for π₯ into a theorem.
(Contributed
by NM, 20-Apr-1994.)
|
β’ (π₯ = π¦ β (π β π)) & β’ π β β’ π |
|
1.4.5 More substitution theorems
|
|
Theorem | hbs1 1938* |
π₯
is not free in [π¦ / π₯]π when π₯ and π¦ are
distinct.
(Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.)
(New usage is discouraged.)
|
β’ ([π¦ / π₯]π β βπ₯[π¦ / π₯]π) |
|
Theorem | nfs1v 1939* |
π₯
is not free in [π¦ / π₯]π when π₯ and π¦ are
distinct.
(Contributed by Mario Carneiro, 11-Aug-2016.)
|
β’ β²π₯[π¦ / π₯]π |
|
Theorem | sbhb 1940* |
Two ways of expressing "π₯ is (effectively) not free in π."
(Contributed by NM, 29-May-2009.)
|
β’ ((π β βπ₯π) β βπ¦(π β [π¦ / π₯]π)) |
|
Theorem | hbsbv 1941* |
This is a version of hbsb 1949 with an extra distinct variable constraint,
on π§ and π₯. (Contributed by Jim
Kingdon, 25-Dec-2017.)
|
β’ (π β βπ§π) β β’ ([π¦ / π₯]π β βπ§[π¦ / π₯]π) |
|
Theorem | nfsbxy 1942* |
Similar to hbsb 1949 but with an extra distinct variable
constraint, on
π₯ and π¦. (Contributed by Jim
Kingdon, 19-Mar-2018.)
|
β’ β²π§π β β’ β²π§[π¦ / π₯]π |
|
Theorem | nfsbxyt 1943* |
Closed form of nfsbxy 1942. (Contributed by Jim Kingdon, 9-May-2018.)
|
β’ (βπ₯β²π§π β β²π§[π¦ / π₯]π) |
|
Theorem | sbco2vlem 1944* |
This is a version of sbco2 1965 where π§ is distinct from π₯ and
from
π¦. It is a lemma on the way to proving
sbco2v 1948 which only
requires that π§ and π₯ be distinct.
(Contributed by Jim Kingdon,
25-Dec-2017.) Remove one disjoint variable condition. (Revised by Jim
Kingdon, 3-Feb-2018.)
|
β’ (π β βπ§π) β β’ ([π¦ / π§][π§ / π₯]π β [π¦ / π₯]π) |
|
Theorem | sbco2vh 1945* |
This is a version of sbco2 1965 where π§ is distinct from π₯.
(Contributed by Jim Kingdon, 12-Feb-2018.)
|
β’ (π β βπ§π) β β’ ([π¦ / π§][π§ / π₯]π β [π¦ / π₯]π) |
|
Theorem | nfsb 1946* |
If π§ is not free in π, it is not free in [π¦ / π₯]π when
π¦ and π§ are distinct.
(Contributed by Mario Carneiro,
11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
|
β’ β²π§π β β’ β²π§[π¦ / π₯]π |
|
Theorem | nfsbv 1947* |
If π§ is not free in π, it is not free in [π¦ / π₯]π when
π§ is distinct from π₯ and
π¦.
Version of nfsb 1946 requiring
more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.)
Remove disjoint variable condition on π₯, π¦. (Revised by Steven
Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by Gino Giotto,
25-Aug-2024.)
|
β’ β²π§π β β’ β²π§[π¦ / π₯]π |
|
Theorem | sbco2v 1948* |
Version of sbco2 1965 with disjoint variable conditions.
(Contributed by
Wolf Lammen, 29-Apr-2023.)
|
β’ β²π§π β β’ ([π¦ / π§][π§ / π₯]π β [π¦ / π₯]π) |
|
Theorem | hbsb 1949* |
If π§ is not free in π, it is not free in [π¦ / π₯]π when
π¦ and π§ are distinct.
(Contributed by NM, 12-Aug-1993.) (Proof
rewritten by Jim Kingdon, 22-Mar-2018.)
|
β’ (π β βπ§π) β β’ ([π¦ / π₯]π β βπ§[π¦ / π₯]π) |
|
Theorem | equsb3lem 1950* |
Lemma for equsb3 1951. (Contributed by NM, 4-Dec-2005.) (Proof
shortened
by Andrew Salmon, 14-Jun-2011.)
|
β’ ([π¦ / π₯]π₯ = π§ β π¦ = π§) |
|
Theorem | equsb3 1951* |
Substitution applied to an atomic wff. (Contributed by Raph Levien and
FL, 4-Dec-2005.)
|
β’ ([π¦ / π₯]π₯ = π§ β π¦ = π§) |
|
Theorem | sbn 1952 |
Negation inside and outside of substitution are equivalent.
(Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon,
3-Feb-2018.)
|
β’ ([π¦ / π₯] Β¬ π β Β¬ [π¦ / π₯]π) |
|
Theorem | sbim 1953 |
Implication inside and outside of substitution are equivalent.
(Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon,
3-Feb-2018.)
|
β’ ([π¦ / π₯](π β π) β ([π¦ / π₯]π β [π¦ / π₯]π)) |
|
Theorem | sbor 1954 |
Logical OR inside and outside of substitution are equivalent.
(Contributed by NM, 29-Sep-2002.) (Proof rewritten by Jim Kingdon,
3-Feb-2018.)
|
β’ ([π¦ / π₯](π β¨ π) β ([π¦ / π₯]π β¨ [π¦ / π₯]π)) |
|
Theorem | sban 1955 |
Conjunction inside and outside of a substitution are equivalent.
(Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon,
3-Feb-2018.)
|
β’ ([π¦ / π₯](π β§ π) β ([π¦ / π₯]π β§ [π¦ / π₯]π)) |
|
Theorem | sbrim 1956 |
Substitution with a variable not free in antecedent affects only the
consequent. (Contributed by NM, 5-Aug-1993.)
|
β’ (π β βπ₯π) β β’ ([π¦ / π₯](π β π) β (π β [π¦ / π₯]π)) |
|
Theorem | sblim 1957 |
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.)
|
β’ β²π₯π β β’ ([π¦ / π₯](π β π) β ([π¦ / π₯]π β π)) |
|
Theorem | sb3an 1958 |
Conjunction inside and outside of a substitution are equivalent.
(Contributed by NM, 14-Dec-2006.)
|
β’ ([π¦ / π₯](π β§ π β§ π) β ([π¦ / π₯]π β§ [π¦ / π₯]π β§ [π¦ / π₯]π)) |
|
Theorem | sbbi 1959 |
Equivalence inside and outside of a substitution are equivalent.
(Contributed by NM, 5-Aug-1993.)
|
β’ ([π¦ / π₯](π β π) β ([π¦ / π₯]π β [π¦ / π₯]π)) |
|
Theorem | sblbis 1960 |
Introduce left biconditional inside of a substitution. (Contributed by
NM, 19-Aug-1993.)
|
β’ ([π¦ / π₯]π β π) β β’ ([π¦ / π₯](π β π) β ([π¦ / π₯]π β π)) |
|
Theorem | sbrbis 1961 |
Introduce right biconditional inside of a substitution. (Contributed by
NM, 18-Aug-1993.)
|
β’ ([π¦ / π₯]π β π) β β’ ([π¦ / π₯](π β π) β (π β [π¦ / π₯]π)) |
|
Theorem | sbrbif 1962 |
Introduce right biconditional inside of a substitution. (Contributed by
NM, 18-Aug-1993.)
|
β’ (π β βπ₯π)
& β’ ([π¦ / π₯]π β π) β β’ ([π¦ / π₯](π β π) β (π β π)) |
|
Theorem | sbco2yz 1963* |
This is a version of sbco2 1965 where π§ is distinct from π¦. It is
a lemma on the way to proving sbco2 1965 which has no distinct variable
constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
|
β’ β²π§π β β’ ([π¦ / π§][π§ / π₯]π β [π¦ / π₯]π) |
|
Theorem | sbco2h 1964 |
A composition law for substitution. (Contributed by NM, 30-Jun-1994.)
(Proof rewritten by Jim Kingdon, 19-Mar-2018.)
|
β’ (π β βπ§π) β β’ ([π¦ / π§][π§ / π₯]π β [π¦ / π₯]π) |
|
Theorem | sbco2 1965 |
A composition law for substitution. (Contributed by NM, 30-Jun-1994.)
(Revised by Mario Carneiro, 6-Oct-2016.)
|
β’ β²π§π β β’ ([π¦ / π§][π§ / π₯]π β [π¦ / π₯]π) |
|
Theorem | sbco2d 1966 |
A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|
β’ (π β βπ₯π)
& β’ (π β βπ§π)
& β’ (π β (π β βπ§π)) β β’ (π β ([π¦ / π§][π§ / π₯]π β [π¦ / π₯]π)) |
|
Theorem | sbco2vd 1967* |
Version of sbco2d 1966 with a distinct variable constraint between
π₯
and π§. (Contributed by Jim Kingdon,
19-Feb-2018.)
|
β’ (π β βπ₯π)
& β’ (π β βπ§π)
& β’ (π β (π β βπ§π)) β β’ (π β ([π¦ / π§][π§ / π₯]π β [π¦ / π₯]π)) |
|
Theorem | sbco 1968 |
A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|
β’ ([π¦ / π₯][π₯ / π¦]π β [π¦ / π₯]π) |
|
Theorem | sbco3v 1969* |
Version of sbco3 1974 with a distinct variable constraint between
π₯
and
π¦. (Contributed by Jim Kingdon,
19-Feb-2018.)
|
β’ ([π§ / π¦][π¦ / π₯]π β [π§ / π₯][π₯ / π¦]π) |
|
Theorem | sbcocom 1970 |
Relationship between composition and commutativity for substitution.
(Contributed by Jim Kingdon, 28-Feb-2018.)
|
β’ ([π§ / π¦][π¦ / π₯]π β [π§ / π¦][π§ / π₯]π) |
|
Theorem | sbcomv 1971* |
Version of sbcom 1975 with a distinct variable constraint between
π₯
and
π§. (Contributed by Jim Kingdon,
28-Feb-2018.)
|
β’ ([π¦ / π§][π¦ / π₯]π β [π¦ / π₯][π¦ / π§]π) |
|
Theorem | sbcomxyyz 1972* |
Version of sbcom 1975 with distinct variable constraints between
π₯
and
π¦, and π¦ and π§.
(Contributed by Jim Kingdon,
21-Mar-2018.)
|
β’ ([π¦ / π§][π¦ / π₯]π β [π¦ / π₯][π¦ / π§]π) |
|
Theorem | sbco3xzyz 1973* |
Version of sbco3 1974 with distinct variable constraints between
π₯
and
π§, and π¦ and π§. Lemma
for proving sbco3 1974. (Contributed
by Jim Kingdon, 22-Mar-2018.)
|
β’ ([π§ / π¦][π¦ / π₯]π β [π§ / π₯][π₯ / π¦]π) |
|
Theorem | sbco3 1974 |
A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
(Proof rewritten by Jim Kingdon, 22-Mar-2018.)
|
β’ ([π§ / π¦][π¦ / π₯]π β [π§ / π₯][π₯ / π¦]π) |
|
Theorem | sbcom 1975 |
A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
(Proof rewritten by Jim Kingdon, 22-Mar-2018.)
|
β’ ([π¦ / π§][π¦ / π₯]π β [π¦ / π₯][π¦ / π§]π) |
|
Theorem | nfsbt 1976* |
Closed form of nfsb 1946. (Contributed by Jim Kingdon, 9-May-2018.)
|
β’ (βπ₯β²π§π β β²π§[π¦ / π₯]π) |
|
Theorem | nfsbd 1977* |
Deduction version of nfsb 1946. (Contributed by NM, 15-Feb-2013.)
|
β’ β²π₯π
& β’ (π β β²π§π) β β’ (π β β²π§[π¦ / π₯]π) |
|
Theorem | sb9v 1978* |
Like sb9 1979 but with a distinct variable constraint
between π₯ and
π¦. (Contributed by Jim Kingdon,
28-Feb-2018.)
|
β’ (βπ₯[π₯ / π¦]π β βπ¦[π¦ / π₯]π) |
|
Theorem | sb9 1979 |
Commutation of quantification and substitution variables. (Contributed
by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
|
β’ (βπ₯[π₯ / π¦]π β βπ¦[π¦ / π₯]π) |
|
Theorem | sb9i 1980 |
Commutation of quantification and substitution variables. (Contributed by
NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
|
β’ (βπ₯[π₯ / π¦]π β βπ¦[π¦ / π₯]π) |
|
Theorem | sbnf2 1981* |
Two ways of expressing "π₯ is (effectively) not free in π."
(Contributed by GΓ©rard Lang, 14-Nov-2013.) (Revised by Mario
Carneiro, 6-Oct-2016.)
|
β’ (β²π₯π β βπ¦βπ§([π¦ / π₯]π β [π§ / π₯]π)) |
|
Theorem | hbsbd 1982* |
Deduction version of hbsb 1949. (Contributed by NM, 15-Feb-2013.) (Proof
rewritten by Jim Kingdon, 23-Mar-2018.)
|
β’ (π β βπ₯π)
& β’ (π β βπ§π)
& β’ (π β (π β βπ§π)) β β’ (π β ([π¦ / π₯]π β βπ§[π¦ / π₯]π)) |
|
Theorem | 2sb5 1983* |
Equivalence for double substitution. (Contributed by NM,
3-Feb-2005.)
|
β’ ([π§ / π₯][π€ / π¦]π β βπ₯βπ¦((π₯ = π§ β§ π¦ = π€) β§ π)) |
|
Theorem | 2sb6 1984* |
Equivalence for double substitution. (Contributed by NM,
3-Feb-2005.)
|
β’ ([π§ / π₯][π€ / π¦]π β βπ₯βπ¦((π₯ = π§ β§ π¦ = π€) β π)) |
|
Theorem | sbcom2v 1985* |
Lemma for proving sbcom2 1987. It is the same as sbcom2 1987 but with
additional distinct variable constraints on π₯ and π¦, and on
π€ and π§. (Contributed by Jim
Kingdon, 19-Feb-2018.)
|
β’ ([π€ / π§][π¦ / π₯]π β [π¦ / π₯][π€ / π§]π) |
|
Theorem | sbcom2v2 1986* |
Lemma for proving sbcom2 1987. It is the same as sbcom2v 1985 but removes
the distinct variable constraint on π₯ and π¦. (Contributed by
Jim Kingdon, 19-Feb-2018.)
|
β’ ([π€ / π§][π¦ / π₯]π β [π¦ / π₯][π€ / π§]π) |
|
Theorem | sbcom2 1987* |
Commutativity law for substitution. Used in proof of Theorem 9.7 of
[Megill] p. 449 (p. 16 of the preprint).
(Contributed by NM,
27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon,
19-Feb-2018.)
|
β’ ([π€ / π§][π¦ / π₯]π β [π¦ / π₯][π€ / π§]π) |
|
Theorem | sb6a 1988* |
Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
|
β’ ([π¦ / π₯]π β βπ₯(π₯ = π¦ β [π₯ / π¦]π)) |
|
Theorem | 2sb5rf 1989* |
Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
|
β’ (π β βπ§π)
& β’ (π β βπ€π) β β’ (π β βπ§βπ€((π§ = π₯ β§ π€ = π¦) β§ [π§ / π₯][π€ / π¦]π)) |
|
Theorem | 2sb6rf 1990* |
Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
|
β’ (π β βπ§π)
& β’ (π β βπ€π) β β’ (π β βπ§βπ€((π§ = π₯ β§ π€ = π¦) β [π§ / π₯][π€ / π¦]π)) |
|
Theorem | dfsb7 1991* |
An alternate definition of proper substitution df-sb 1763. By introducing
a dummy variable π§ in the definiens, we are able to
eliminate any
distinct variable restrictions among the variables π₯, π¦, and
π of the definiendum. No distinct
variable conflicts arise because
π§ effectively insulates π₯ from
π¦.
To achieve this, we use
a chain of two substitutions in the form of sb5 1887,
first π§ for
π₯ then π¦ for π§.
Compare Definition 2.1'' of [Quine] p. 17.
Theorem sb7f 1992 provides a version where π and π§ don't
have to
be distinct. (Contributed by NM, 28-Jan-2004.)
|
β’ ([π¦ / π₯]π β βπ§(π§ = π¦ β§ βπ₯(π₯ = π§ β§ π))) |
|
Theorem | sb7f 1992* |
This version of dfsb7 1991 does not require that π and π§ be
disjoint. This permits it to be used as a definition for substitution
in a formalization that omits the logically redundant axiom ax-17 1526,
i.e., that does not have the concept of a variable not occurring in a
formula. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew
Salmon, 25-May-2011.)
|
β’ (π β βπ§π) β β’ ([π¦ / π₯]π β βπ§(π§ = π¦ β§ βπ₯(π₯ = π§ β§ π))) |
|
Theorem | sb7af 1993* |
An alternate definition of proper substitution df-sb 1763. Similar to
dfsb7a 1994 but does not require that π and π§ be
distinct.
Similar to sb7f 1992 in that it involves a dummy variable π§, but
expressed in terms of β rather than β. (Contributed by Jim
Kingdon, 5-Feb-2018.)
|
β’ β²π§π β β’ ([π¦ / π₯]π β βπ§(π§ = π¦ β βπ₯(π₯ = π§ β π))) |
|
Theorem | dfsb7a 1994* |
An alternate definition of proper substitution df-sb 1763. Similar to
dfsb7 1991 in that it involves a dummy variable π§, but
expressed in
terms of β rather than β. For a version which only requires
β²π§π rather than π§ and π being
distinct, see sb7af 1993.
(Contributed by Jim Kingdon, 5-Feb-2018.)
|
β’ ([π¦ / π₯]π β βπ§(π§ = π¦ β βπ₯(π₯ = π§ β π))) |
|
Theorem | sb10f 1995* |
Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed.,
1979), p. 328. In traditional predicate calculus, this is a sole axiom
for identity from which the usual ones can be derived. (Contributed by
NM, 9-May-2005.)
|
β’ (π β βπ₯π) β β’ ([π¦ / π§]π β βπ₯(π₯ = π¦ β§ [π₯ / π§]π)) |
|
Theorem | sbid2v 1996* |
An identity law for substitution. Used in proof of Theorem 9.7 of
[Megill] p. 449 (p. 16 of the preprint).
(Contributed by NM,
5-Aug-1993.)
|
β’ ([π¦ / π₯][π₯ / π¦]π β π) |
|
Theorem | sbelx 1997* |
Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
|
β’ (π β βπ₯(π₯ = π¦ β§ [π₯ / π¦]π)) |
|
Theorem | sbel2x 1998* |
Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
|
β’ (π β βπ₯βπ¦((π₯ = π§ β§ π¦ = π€) β§ [π¦ / π€][π₯ / π§]π)) |
|
Theorem | sbalyz 1999* |
Move universal quantifier in and out of substitution. Identical to
sbal 2000 except that it has an additional distinct
variable constraint on
π¦ and π§. (Contributed by Jim
Kingdon, 29-Dec-2017.)
|
β’ ([π§ / π¦]βπ₯π β βπ₯[π§ / π¦]π) |
|
Theorem | sbal 2000* |
Move universal quantifier in and out of substitution. (Contributed by
NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
|
β’ ([π§ / π¦]βπ₯π β βπ₯[π§ / π¦]π) |