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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdrsb2 2001 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.)
 |-  ( A. x  x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
 
Theoremsbn 2002 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
 
Theoremsbi1 2003 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbi2 2004 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
 
Theoremsbim 2005 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbor 2006 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
 |-  ( [ y  /  x ] ( ph  \/  ps )  <->  ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) )
 
Theoremsbrim 2007 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.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  x ] ps ) )
 
Theoremsblim 2008 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 ) )
 
Theoremsban 2009 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  /\  ps ) 
 <->  ( [ y  /  x ] ph  /\  [
 y  /  x ] ps ) )
 
Theoremsb3an 2010 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 ) )
 
Theoremsbbi 2011 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 ) )
 
Theoremsblbis 2012 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 ) )
 
Theoremsbrbis 2013 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 ) )
 
Theoremsbrbif 2014 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ch   &    |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theoremspsbe 2015 A specialization theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
Theoremspsbim 2016 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
 )
 
Theoremspsbbi 2017 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
Theoremsbbid 2018 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  [ y  /  x ] ch ) )
 
Theoremsbequ8 2019 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremnfsb4t 2020 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2021). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4 2021 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.)
 |- 
 F/ z ph   =>    |-  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
 
Theoremdvelimdf 2022 Deduction form of dvelimf 1937. This version may be useful if we want to avoid ax-17 1603 and use ax-16 2083 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ z ch )   &    |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( -.  A. x  x  =  y 
 ->  F/ x ch )
 )
 
Theoremsbco 2023 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
Theoremsbid2 2024 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
Theoremsbidm 2025 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsbco2 2026 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 )
 
Theoremsbco2d 2027 A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  ( [ y  /  z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
 
Theoremsbco3 2028 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
Theoremsbcom 2029 A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb5rf 2030 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( ph  <->  E. y ( y  =  x  /\  [
 y  /  x ] ph ) )
 
Theoremsb6rf 2031 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
 )
 
Theoremsb8 2032 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8e 2033 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
Theoremsb9i 2034 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  ->  A. y [ y  /  x ] ph )
 
Theoremsb9 2035 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
 
Theoremax11v 2036* This is a version of ax-11o 2080 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 1932 for the rederivation of ax-11o 2080 from this theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremsb56 2037* 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 1630. (Contributed by NM, 14-Apr-2008.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb6 2038* 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 ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb5 2039* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremequsb3lem 2040* Lemma for equsb3 2041. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  z  <-> 
 x  =  z )
 
Theoremequsb3 2041* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
 |-  ( [ x  /  y ] y  =  z  <-> 
 x  =  z )
 
Theoremelsb3 2042* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  e.  z  <->  x  e.  z )
 
Theoremelsb4 2043* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
 
Theoremhbs1 2044*  x is not free in  [ y  /  x ] ph when  x and  y are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 
Theoremnfs1v 2045*  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
 
Theoremsbhb 2046* 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 ) )
 
Theoremsbnf2 2047* 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 ) )
 
Theoremnfsb 2048* 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.)
 |- 
 F/ z ph   =>    |- 
 F/ z [ y  /  x ] ph
 
Theoremhbsb 2049* 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.)
 |-  ( ph  ->  A. z ph )   =>    |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
 
Theoremnfsbd 2050* Deduction version of nfsb 2048. (Contributed by NM, 15-Feb-2013.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  F/ z [ y  /  x ] ps )
 
Theorem2sb5 2051* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  ph )
 )
 
Theorem2sb6 2052* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x A. y
 ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theoremsbcom2 2053* 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.)
 |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theorempm11.07 2054* Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theoremsb6a 2055* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  [ x  /  y ] ph )
 )
 
Theorem2sb5rf 2056* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   =>    |-  ( ph 
 <-> 
 E. z E. w ( ( z  =  x  /\  w  =  y )  /\  [
 z  /  x ] [ w  /  y ] ph ) )
 
Theorem2sb6rf 2057* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   =>    |-  ( ph 
 <-> 
 A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph )
 )
 
Theoremdfsb7 2058* An alternate definition of proper substitution df-sb 1630. By introducing a dummy variable  z in the definiens, we are able to eliminate any distinct variable restrictions among the variables  x,  y, and  ph of the definiendum. No distinct variable conflicts arise because  z effectively insulates  x from  y. To achieve this, we use a chain of two substitutions in the form of sb5 2039, first  z for  x then  y for  z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2270. Theorem sb7h 2060 provides a version where  ph and  z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
 |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb7f 2059* This version of dfsb7 2058 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1603 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1630 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   =>    |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb7h 2060* This version of dfsb7 2058 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1603 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1630 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A. z ph )   =>    |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb10f 2061* 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.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  z ] ph  <->  E. x ( x  =  y  /\  [ x  /  z ] ph ) )
 
Theoremsbid2v 2062* 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.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
Theoremsbelx 2063* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  E. x ( x  =  y  /\  [ x  /  y ] ph ) )
 
Theoremsbel2x 2064* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  [
 y  /  w ] [ x  /  z ] ph ) )
 
Theoremsbal1 2065* A theorem used in elimination of disjoint variable restriction on  x and  y by replacing it with a distinctor  -.  A. x x  =  z. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  z  ->  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 )
 
Theoremsbal 2066* Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 
Theoremsbex 2067* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
 |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
 
Theoremsbalv 2068* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
 
Theoremexsb 2069* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph )
 )
 
TheoremexsbOLD 2070* An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.)
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph )
 )
 
Theorem2exsb 2071* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
 |-  ( E. x E. y ph  <->  E. z E. w A. x A. y ( ( x  =  z 
 /\  y  =  w )  ->  ph ) )
 
TheoremdvelimALT 2072* Version of dvelim 1956 that doesn't use ax-10 2079. (See dvelimh 1904 for a version that doesn't use ax-11 1715.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremsbal2 2073* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 )
 
1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)

The "metalogical completeness theorem", Theorem 9.7 of [Megill] p. 448, uses a different but (logically and metalogically) equivalent set of axiom schemes for its proof. In order to show that our axiomatization is also metalogically complete, we derive the axiom schemes of that paper in this section (or mention where they are derived, if they have already been derived as therorems above). Additionally, we re-derive our axiomatization from the one in the paper, showing that the two systems are equivalent.

The 14 predicate calculus axioms used by the paper are ax-5o 2075, ax-4 2074, ax-7 1708, ax-6o 2076, ax-8 1643, ax-12o 2081, ax-9o 2077, ax-10o 2078, ax-13 1686, ax-14 1688, ax-15 2082, ax-11o 2080, ax-16 2083, and ax-17 1603. Like ours, it includes the rule of generalization (ax-gen 1533).

The ones we need to prove from our axioms are ax-5o 2075, ax-4 2074, ax-6o 2076, ax-12o 2081, ax-9o 2077, ax-10o 2078, ax-15 2082, ax-11o 2080, and ax-16 2083. The theorems showing the derivations of those axioms, which have all been proved earlier, are ax5o 1717, ax4 2084 (also called sp 1716), ax6o 1723, ax12o 1875, ax9o 1890, ax10o 1892, ax15 1961, ax11o 1934, ax16 1985, and ax10 1884. In addition, ax-10 2079 was an intermediate axiom we adopted at one time, and we show its proof in this section as ax10from10o 2116.

This section also includes a few miscellaneous legacy theorems such as hbequid 2099 use the older axioms.

Note: The axioms and theorems in this section should not be used outside of this section. Inside this section, we may use the external axioms ax-gen 1533, ax-17 1603, ax-8 1643, ax-9 1635, ax-13 1686, and ax-14 1688 since they are common to both our current and the older axiomatizations. (These are the ones that were never revised.)

The following newer axioms may NOT be used in this section until we have proved them from the older axioms: ax-5 1544, ax-6 1703, ax-9 1635, ax-11 1715, and ax-12 1866. However, once we have rederived an axiom (e.g. theorem ax5 2085 for axiom ax-5 1544), we may make use of theorems outside of this section that make use of the rederived axiom (e.g. we may use theorem alimi 1546, which uses ax-5 1544, after proving ax5 2085).

 
1.6.1  Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16

These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems ax5o , sp 1716, ax6o 1723, ax9o 1890, ax10o 1892, ax10 1884, ax11o 1934, ax12o 1875, ax15 1961, and ax16 1985.

 
Axiomax-4 2074 Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all  x, it is true for any specific  x (that would typically occur as a free variable in the wff substituted for  ph). (A free variable is one that does not occur in the scope of a quantifier:  x and  y are both free in  x  =  y, but only  x is free in  A. y x  =  y.) This is one of the axioms of what we call "pure" predicate calculus (ax-4 2074 through ax-7 1708 plus rule ax-gen 1533). Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1533. Conditional forms of the converse are given by ax-12 1866, ax-15 2082, ax-16 2083, and ax-17 1603.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from  x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1964.

An interesting alternate axiomatization uses ax467 2108 and ax-5o 2075 in place of ax-4 2074, ax-5 1544, ax-6 1703, and ax-7 1708.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1716. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x ph  -> 
 ph )
 
Axiomax-5o 2075 Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying  ps. Notice that  x must not be a free variable in the antecedent of the quantified implication, and we express this by binding  ph to "protect" the axiom from a  ph containing a free  x. One of the 4 axioms of "pure" predicate calculus. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax5o 1717. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x (
 A. x ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Axiomax-6o 2076 Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. One of the 4 axioms of pure predicate calculus. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use ax467 2108 in place of ax-4 2074, ax-6o 2076, and ax-7 1708.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax6o 1723. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Axiomax-9o 2077 A variant of ax9 1889. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as theorem ax9o 1890. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
 
Axiomax-10o 2078 Axiom ax-10o 2078 ("o" for "old") was the original version of ax-10 2079, before it was discovered (in May 2008) that the shorter ax-10 2079 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as theorem ax10o 1892. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
Axiomax-10 2079 Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-10o 2078 ("o" for "old") and was replaced with this shorter ax-10 2079 in May 2008. The old axiom is proved from this one as theorem ax10o 1892. Conversely, this axiom is proved from ax-10o 2078 as theorem ax10from10o 2116.

This axiom was proved redundant in July 2015. See theorem ax10 1884.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 1884. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Axiomax-11o 2080 Axiom ax-11o 2080 ("o" for "old") was the original version of ax-11 1715, before it was discovered (in Jan. 2007) that the shorter ax-11 1715 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of " -.  A. x x  =  y  ->..." as informally meaning "if  x and  y are distinct variables then..." The antecedent becomes false if the same variable is substituted for  x and  y, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form  -.  A. x x  =  y a "distinctor."

Interestingly, if the wff expression substituted for  ph contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 2080 (from which the ax-11 1715 instance follows by theorem ax11 2094.) The proof is by induction on formula length, using ax11eq 2132 and ax11el 2133 for the basis steps and ax11indn 2134, ax11indi 2135, and ax11inda 2139 for the induction steps. (This paragraph is true provided we use ax-10o 2078 in place of ax-10 2079.)

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1934. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Axiomax-12o 2081 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever  z is distinct from  x and  y, and  x  =  y is true, then  x  =  y quantified with  z is also true. In other words,  z is irrelevant to the truth of 
x  =  y. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax12o 1875. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
 
Axiomax-15 2082 Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-17 1603; see theorem ax15 1961. Alternately, ax-17 1603 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1603. We retain ax-15 2082 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1603, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax15 1961. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y
 ) ) )
 
Axiomax-16 2083* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1603 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 4201), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1603; see theorem ax16 1985. Alternately, ax-17 1603 becomes logically redundant in the presence of this axiom, but without ax-17 1603 we lose the more powerful metalogic that results from being able to express the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). We retain ax-16 2083 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1603, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1985. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
1.6.2  Rederive new axioms from old: theorems ax5 , ax6 , ax9from9o , ax11 , ax12

Theorems ax11 2094 and ax12 2095 require some intermediate theorems that are included in this section.

 
Theoremax4 2084 This theorem repeats sp 1716 under the name ax4 2084, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-4 2074. It is preferred that references to this theorem use the name sp 1716. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremax5 2085 Rederivation of axiom ax-5 1544 from ax-5o 2075 and other older axioms. See ax5o 1717 for the derivation of ax-5o 2075 from ax-5 1544. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremax6 2086 Rederivation of axiom ax-6 1703 from ax-6o 2076 and other older axioms. See ax6o 1723 for the derivation of ax-6o 2076 from ax-6 1703. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremax9from9o 2087 Rederivation of axiom ax-9 1635 from ax-9o 2077 and other older axioms. See ax9o 1890 for the derivation of ax-9o 2077 from ax-9 1635. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  A. x  -.  x  =  y
 
Theoremhba1-o 2088  x is not free in  A. x ph. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( A. x ph  ->  A. x A. x ph )
 
Theorema5i-o 2089 Inference version of ax-5o 2075. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theoremaecom-o 2090 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 1886 using ax-10o 2078. Unlike ax10from10o 2116, this version does not require ax-17 1603. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremaecoms-o 2091 A commutation rule for identical variable specifiers. Version of aecoms 1887 using ax-10o . (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremhbae-o 2092 All variables are effectively bound in an identical variable specifier. Version of hbae 1893 using ax-10o 2078. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremdral1-o 2093 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 1905 using ax-10o 2078. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. y ps )
 )
 
Theoremax11 2094 Rederivation of axiom ax-11 1715 from ax-11o 2080, ax-10o 2078, and other older axioms. The proof does not require ax-16 2083 or ax-17 1603. See theorem ax11o 1934 for the derivation of ax-11o 2080 from ax-11 1715.

An open problem is whether we can prove this using ax-10 2079 instead of ax-10o 2078.

This proof uses newer axioms ax-5 1544 and ax-9 1635, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2075 and ax-9o 2077. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax12 2095 Derive ax-12 1866 from ax-12o 2081 and other older axioms.

This proof uses newer axioms ax-5 1544 and ax-9 1635, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2075 and ax-9o 2077. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
1.6.3  Legacy theorems using obsolete axioms

These theorems were mostly intended to study properties of the older axiom schemes and are not useful outside of this section. They should not be used outside of this section. They may be deleted when they are deemed to no longer be of interest.

 
Theoremax17o 2096* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(This theorem simply repeats ax-17 1603 so that we can include the following note, which applies only to the obsolete axiomatization.)

This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1533, ax-5o 2075, ax-4 2074, ax-7 1708, ax-6o 2076, ax-8 1643, ax-12o 2081, ax-9o 2077, ax-10o 2078, ax-13 1686, ax-14 1688, ax-15 2082, ax-11o 2080, and ax-16 2083: in that system, we can derive any instance of ax-17 1603 not containing wff variables by induction on formula length, using ax17eq 2122 and ax17el 2128 for the basis together hbn 1720, hbal 1710, and hbim 1725. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification discouraged.)

 |-  ( ph  ->  A. x ph )
 
Theoremequid1 2097 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and thus requires no dummy variables. A simpler proof, similar to Tarki's, is possible if we make use of ax-17 1603; see the proof of equid 1644. See equid1ALT 2115 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  x  =  x
 
Theoremsps-o 2098 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theoremhbequid 2099 Bound-variable hypothesis builder for  x  =  x. This theorem tells us that any variable, including  x, is effectively not free in  x  =  x, even though  x is technically free according to the traditional definition of free variable. (The proof does not use ax-9o 2077.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  x 
 ->  A. y  x  =  x )
 
Theoremnfequid-o 2100 Bound-variable hypothesis builder for  x  =  x. This theorem tells us that any variable, including  x, is effectively not free in  x  =  x, even though  x is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1544, ax-8 1643, ax-12o 2081, and ax-gen 1533. This shows that this can be proved without ax9 1889, even though the theorem equid 1644 cannot be. A shorter proof using ax9 1889 is obtainable from equid 1644 and hbth 1539.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1636, which is used for the derivation of ax12o 1875, unless we consider ax-12o 2081 the starting axiom rather than ax-12 1866. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y  x  =  x
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