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Theorem List for New Foundations Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremequvin 2001* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
(x = yz(x = z z = y))

Theoremcbvalv 2002* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
(x = y → (φψ))       (xφyψ)

Theoremcbvexv 2003* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
(x = y → (φψ))       (xφyψ)

Theoremcbval2 2004* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
zφ    &   wφ    &   xψ    &   yψ    &   ((x = z y = w) → (φψ))       (xyφzwψ)

Theoremcbvex2 2005* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
zφ    &   wφ    &   xψ    &   yψ    &   ((x = z y = w) → (φψ))       (xyφzwψ)

Theoremcbval2v 2006* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
((x = z y = w) → (φψ))       (xyφzwψ)

Theoremcbvex2v 2007* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
((x = z y = w) → (φψ))       (xyφzwψ)

Theoremcbvald 2008* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
yφ    &   (φ → Ⅎyψ)    &   (φ → (x = y → (ψχ)))       (φ → (xψyχ))

Theoremcbvexd 2009* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
yφ    &   (φ → Ⅎyψ)    &   (φ → (x = y → (ψχ)))       (φ → (xψyχ))

Theoremcbvaldva 2010* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))       (φ → (xψyχ))

Theoremcbvexdva 2011* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))       (φ → (xψyχ))

Theoremcbvex4v 2012* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
((x = v y = u) → (φψ))    &   ((z = f w = g) → (ψχ))       (xyzwφvufgχ)

Theoremchvarv 2013* Implicit substitution of y for x into a theorem. (Contributed by NM, 20-Apr-1994.)
(x = y → (φψ))    &   φ       ψ

Theoremcleljust 2014* When the class variables in definition df-clel 2349 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1711 with the class variables in wcel 1710. Note: This proof is referenced on the Metamath Proof Explorer Home Page and shouldn't be changed. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.)
(x yz(z = x z y))

TheoremcleljustALT 2015* When the class variables in definition df-clel 2349 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1711 with the class variables in wcel 1710. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.)
(x yz(z = x z y))

Theoremdvelim 2016* This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" ¬ xx = y as an antecedent. φ normally has z free and can be read φ(z), and ψ substitutes y for z and can be read φ(y). We don't require that x and y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with xz, conjoin them, and apply dvelimdf 2082.

Other variants of this theorem are dvelimh 1964 (with no distinct variable restrictions), dvelimhw 1849 (that avoids ax-12 1925), and dvelimALT 2133 (that avoids ax-10 2140). (Contributed by NM, 23-Nov-1994.)

(φxφ)    &   (z = y → (φψ))       x x = y → (ψxψ))

Theoremdvelimnf 2017* Version of dvelim 2016 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)
xφ    &   (z = y → (φψ))       x x = y → Ⅎxψ)

Theoremdveeq1 2018* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
x x = y → (y = zx y = z))

Theoremdveel1 2019* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
x x = y → (y zx y z))

Theoremdveel2 2020* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
x x = y → (z yx z y))

Theoremax15 2021 Axiom ax-15 2143 is redundant if we assume ax-17 1616. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that w is a dummy variable introduced in the proof. On the web page, it is implicitly assumed to be distinct from all other variables. (This is made explicit in the database file set.mm). Its purpose is to satisfy the distinct variable requirements of dveel2 2020 and ax-17 1616. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.)

z z = x → (¬ z z = y → (x yz x y)))

Theoremdrsb1 2022 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, 5-Aug-1993.)
(x x = y → ([z / x]φ ↔ [z / y]φ))

Theoremsb2 2023 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
(x(x = yφ) → [y / x]φ)

Theoremstdpc4 2024 The specialization axiom of standard predicate calculus. It states that if a statement φ holds for all x, then it also holds for the specific case of y (properly) substituted for x. Translated to traditional notation, it can be read: "xφ(x) → φ(y), provided that y is free for x in φ(x)." Axiom 4 of [Mendelson] p. 69. See also spsbc 3058 and rspsbc 3124. (Contributed by NM, 5-Aug-1993.)
(xφ → [y / x]φ)

Theoremsbft 2025 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.)
(Ⅎxφ → ([y / x]φφ))

Theoremsbf 2026 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
xφ       ([y / x]φφ)

Theoremsbh 2027 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)
(φxφ)       ([y / x]φφ)

Theoremsbf2 2028 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
([y / x]xφxφ)

Theoremsb6x 2029 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
xφ       ([y / x]φx(x = yφ))

Theoremnfs1f 2030 If x is not free in φ, it is not free in [y / x]φ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       x[y / x]φ

Theoremsbequ5 2031 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.)
([w / z]x x = yx x = y)

Theoremsbequ6 2032 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)
([w / z] ¬ x x = y ↔ ¬ x x = y)

Theoremsbt 2033 A substitution into a theorem remains true. (See chvar 1986 and chvarv 2013 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
φ       [y / x]φ

Theoremequsb1 2034 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
[y / x]x = y

Theoremequsb2 2035 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
[y / x]y = x

Theoremsbied 2036 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2038). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
xφ    &   (φ → Ⅎxχ)    &   (φ → (x = y → (ψχ)))       (φ → ([y / x]ψχ))

Theoremsbiedv 2037* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2038). (Contributed by NM, 7-Jan-2017.)
((φ x = y) → (ψχ))       (φ → ([y / x]ψχ))

Theoremsbie 2038 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
xψ    &   (x = y → (φψ))       ([y / x]φψ)

Theoremsb6f 2039 Equivalence for substitution when y is not free in φ. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
yφ       ([y / x]φx(x = yφ))

Theoremsb5f 2040 Equivalence for substitution when y is not free in φ. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
yφ       ([y / x]φx(x = y φ))

Theoremhbsb2a 2041 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([y / x]yφx[y / x]φ)

Theoremhbsb2e 2042 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([y / x]φx[y / x]yφ)

Theoremhbsb3 2043 If y is not free in φ, x is not free in [y / x]φ. (Contributed by NM, 5-Aug-1993.)
(φyφ)       ([y / x]φx[y / x]φ)

Theoremnfs1 2044 If y is not free in φ, x is not free in [y / x]φ. (Contributed by Mario Carneiro, 11-Aug-2016.)
yφ       x[y / x]φ

Theoremax16 2045* Proof of older axiom ax-16 2144. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
(x x = y → (φxφ))

Theoremax16i 2046* Inference with ax16 2045 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
(x = z → (φψ))    &   (ψxψ)       (x x = y → (φxφ))

Theoremax16ALT 2047* Alternate proof of ax16 2045. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(x x = y → (φxφ))

Theoremax16ALT2 2048* Alternate proof of ax16 2045. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(x x = y → (φxφ))

Theorema16gALT 2049* A generalization of axiom ax-16 2144. Alternate proof of a16g 1945 that uses df-sb 1649. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(x x = y → (φzφ))

Theorema16gb 2050* A generalization of axiom ax-16 2144. (Contributed by NM, 5-Aug-1993.)
(x x = y → (φzφ))

Theorema16nf 2051* If dtru in set.mm is false, then there is only one element in the universe, so everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.)
(x x = y → Ⅎzφ)

Theoremsb3 2052 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
x x = y → (x(x = y φ) → [y / x]φ))

Theoremsb4 2053 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
x x = y → ([y / x]φx(x = yφ)))

Theoremsb4b 2054 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
x x = y → ([y / x]φx(x = yφ)))

Theoremdfsb2 2055 An alternate definition of proper substitution that, like df-sb 1649, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)
([y / x]φ ↔ ((x = y φ) x(x = yφ)))

Theoremdfsb3 2056 An alternate definition of proper substitution df-sb 1649 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)
([y / x]φ ↔ ((x = y → ¬ φ) → x(x = yφ)))

Theoremhbsb2 2057 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
x x = y → ([y / x]φx[y / x]φ))

Theoremnfsb2 2058 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
x x = y → Ⅎx[y / x]φ)

Theoremsbequi 2059 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
(x = y → ([x / z]φ → [y / z]φ))

Theoremsbequ 2060 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.)
(x = y → ([x / z]φ ↔ [y / z]φ))

Theoremdrsb2 2061 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.)
(x x = y → ([x / z]φ ↔ [y / z]φ))

Theoremsbn 2062 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
([y / x] ¬ φ ↔ ¬ [y / x]φ)

Theoremsbi1 2063 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)
([y / x](φψ) → ([y / x]φ → [y / x]ψ))

Theoremsbi2 2064 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
(([y / x]φ → [y / x]ψ) → [y / x](φψ))

Theoremsbim 2065 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))

Theoremsbor 2066 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))

Theoremsbrim 2067 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.)
xφ       ([y / x](φψ) ↔ (φ → [y / x]ψ))

Theoremsblim 2068 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.)
xψ       ([y / x](φψ) ↔ ([y / x]φψ))

Theoremsban 2069 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))

Theoremsb3an 2070 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
([y / x](φ ψ χ) ↔ ([y / x]φ [y / x]ψ [y / x]χ))

Theoremsbbi 2071 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
([y / x](φψ) ↔ ([y / x]φ ↔ [y / x]ψ))

Theoremsblbis 2072 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
([y / x]φψ)       ([y / x](χφ) ↔ ([y / x]χψ))

Theoremsbrbis 2073 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
([y / x]φψ)       ([y / x](φχ) ↔ (ψ ↔ [y / x]χ))

Theoremsbrbif 2074 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
xχ    &   ([y / x]φψ)       ([y / x](φχ) ↔ (ψχ))

Theoremspsbe 2075 A specialization theorem. (Contributed by NM, 5-Aug-1993.)
([y / x]φxφ)

Theoremspsbim 2076 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(x(φψ) → ([y / x]φ → [y / x]ψ))

Theoremspsbbi 2077 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)
(x(φψ) → ([y / x]φ ↔ [y / x]ψ))

Theoremsbbid 2078 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
xφ    &   (φ → (ψχ))       (φ → ([y / x]ψ ↔ [y / x]χ))

Theoremsbequ8 2079 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)
([y / x]φ ↔ [y / x](x = yφ))

Theoremnfsb4t 2080 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2081). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.)
(xzφ → (¬ z z = y → Ⅎz[y / x]φ))

Theoremnfsb4 2081 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.)
zφ       z z = y → Ⅎz[y / x]φ)

Theoremdvelimdf 2082 Deduction form of dvelimf 1997. This version may be useful if we want to avoid ax-17 1616 and use ax-16 2144 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)
xφ    &   zφ    &   (φ → Ⅎxψ)    &   (φ → Ⅎzχ)    &   (φ → (z = y → (ψχ)))       (φ → (¬ x x = y → Ⅎxχ))

Theoremsbco 2083 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
([y / x][x / y]φ ↔ [y / x]φ)

Theoremsbid2 2084 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
xφ       ([y / x][x / y]φφ)

Theoremsbidm 2085 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
([y / x][y / x]φ ↔ [y / x]φ)

Theoremsbco2 2086 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
zφ       ([y / z][z / x]φ ↔ [y / x]φ)

Theoremsbco2d 2087 A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
xφ    &   zφ    &   (φ → Ⅎzψ)       (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))

Theoremsbco3 2088 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
([z / y][y / x]φ ↔ [z / x][x / y]φ)

Theoremsbcom 2089 A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
([y / z][y / x]φ ↔ [y / x][y / z]φ)

Theoremsb5rf 2090 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
yφ       (φy(y = x [y / x]φ))

Theoremsb6rf 2091 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
yφ       (φy(y = x → [y / x]φ))

Theoremsb8 2092 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
yφ       (xφy[y / x]φ)

Theoremsb8e 2093 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
yφ       (xφy[y / x]φ)

Theoremsb9i 2094 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
(x[x / y]φy[y / x]φ)

Theoremsb9 2095 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
(x[x / y]φy[y / x]φ)

Theoremax11v 2096* This is a version of ax-11o 2141 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 1992 for the rederivation of ax-11o 2141 from this theorem. (Contributed by NM, 5-Aug-1993.)
(x = y → (φx(x = yφ)))

Theoremax11vALT 2097* Alternate proof of ax11v 2096 that avoids theorem ax16 2045 and is proved directly from ax-11 1746 rather than via ax11o 1994. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(x = y → (φx(x = yφ)))

Theoremsb56 2098* Two equivalent ways of expressing the proper substitution of y for x in φ, when x and y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1649. (Contributed by NM, 14-Apr-2008.)
(x(x = y φ) ↔ x(x = yφ))

Theoremsb6 2099* 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.)
([y / x]φx(x = yφ))

Theoremsb5 2100* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)
([y / x]φx(x = y φ))

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