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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspsbe 2101 A specialization theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
Theoremspsbim 2102 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 2103 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
Theoremsbbid 2104 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 2105 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremnfsb4t 2106 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2107). (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 2107 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 2108 Deduction form of dvelimf 2023. This version may be useful if we want to avoid ax-17 1623 and use ax-16 2171 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 2109 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
Theoremsbid2 2110 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 2111 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 2112 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 2113 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 2114 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
Theoremsbcom 2115 A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb5rf 2116 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 2117 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 2118 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 )
 
Theoremsb8e 2119 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 )
 
Theoremsb9i 2120 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  ->  A. y [ y  /  x ] ph )
 
Theoremsb9 2121 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
 
Theoremax11v 2122* This is a version of ax-11o 2168 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2018 for the rederivation of ax-11o 2168 from this theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremax11vALT 2123* Alternate proof of ax11v 2122 that avoids theorem ax16 2071 and is proved directly from ax-11 1753 rather than via ax11o 2020. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremsb56 2124* 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 1656. (Contributed by NM, 14-Apr-2008.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb6 2125* 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 2126* 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 2127* Lemma for equsb3 2128. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  z  <-> 
 x  =  z )
 
Theoremequsb3 2128* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
 |-  ( [ x  /  y ] y  =  z  <-> 
 x  =  z )
 
Theoremelsb3 2129* 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 2130* 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 2131*  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 2132*  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 2133* 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 2134* 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 2135* 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 2136* 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 2137* Deduction version of nfsb 2135. (Contributed by NM, 15-Feb-2013.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  F/ z [ y  /  x ] ps )
 
Theorem2sb5 2138* 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 2139* 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 2140* 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 2141* (Probably not) Axiom *11.07 in [WhiteheadRussell] p. 159. The original confusingly reads: *11.07 "Whatever possible argument  x may be,  ph ( x ,  y ) is true whatever possible argument  y may be" implies the corresponding statement with  x and  y interchanged except in " ph ( x ,  y )". This theorem will be deleted after 22-Feb-2018 if no one is able to determine the correct interpretation. See https://groups.google.com/d/msg/metamath/iS0fOvSemC8/YzrRyX70AgAJ. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.) (New usage is discouraged.)
 |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theorempm11.07OLD 2142* Obsolete proof of pm11.07 2141 as of 22-Jan-2018. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theoremsb6a 2143* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  [ x  /  y ] ph )
 )
 
Theorem2sb5rf 2144* 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 2145* 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 2146* An alternate definition of proper substitution df-sb 1656. 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 2126, 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 2367. Theorem sb7h 2148 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 2147* This version of dfsb7 2146 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 1623 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1656 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 2148* This version of dfsb7 2146 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 1623 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1656 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 2149* 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 2150* 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 2151* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  E. x ( x  =  y  /\  [ x  /  y ] ph ) )
 
Theoremsbel2x 2152* 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 2153* 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 2154* 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 2155* 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 2156* 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 2157* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph )
 )
 
TheoremexsbOLD 2158* An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph )
 )
 
Theorem2exsb 2159* 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 2160* Version of dvelim 2042 that doesn't use ax-10 2167. (See dvelimh 1990 for a version that doesn't use ax-11 1753.) (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 2161* 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 2163, ax-4 2162, ax-7 1741, ax-6o 2164, ax-8 1682, ax-12o 2169, ax-9o 2165, ax-10o 2166, ax-13 1719, ax-14 1721, ax-15 2170, ax-11o 2168, ax-16 2171, and ax-17 1623. Like ours, it includes the rule of generalization (ax-gen 1552).

The ones we need to prove from our axioms are ax-5o 2163, ax-4 2162, ax-6o 2164, ax-12o 2169, ax-9o 2165, ax-10o 2166, ax-15 2170, ax-11o 2168, and ax-16 2171. The theorems showing the derivations of those axioms, which have all been proved earlier, are ax5o 1757, ax4 2172 (also called sp 1755), ax6o 1758, ax12o 1956, ax9o 1943, ax10o 1983, ax15 2047, ax11o 2020, ax16 2071, and ax10 1976. In addition, ax-10 2167 was an intermediate axiom we adopted at one time, and we show its proof in this section as ax10from10o 2204.

This section also includes a few miscellaneous legacy theorems such as hbequid 2187 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 1552, ax-17 1623, ax-8 1682, ax-9 1661, ax-13 1719, and ax-14 1721 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 1563, ax-6 1736, ax-9 1661, ax-11 1753, and ax-12 1939. However, once we have rederived an axiom (e.g. theorem ax5 2173 for axiom ax-5 1563), we may make use of theorems outside of this section that make use of the rederived axiom (e.g. we may use theorem alimi 1565, which uses ax-5 1563, after proving ax5 2173).

 
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 1755, ax6o 1758, ax9o 1943, ax10o 1983, ax10 1976, ax11o 2020, ax12o 1956, ax15 2047, and ax16 2071.

 
Axiomax-4 2162 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.) 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 1552. Conditional forms of the converse are given by ax-12 1939, ax-15 2170, ax-16 2171, and ax-17 1623.

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 2050.

An interesting alternate axiomatization uses ax467 2196 and ax-5o 2163 in place of ax-4 2162, ax-5 1563, ax-6 1736, and ax-7 1741.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1755. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x ph  -> 
 ph )
 
Axiomax-5o 2163 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. 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 1757. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x (
 A. x ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Axiomax-6o 2164 Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use ax467 2196 in place of ax-4 2162, ax-6o 2164, and ax-7 1741.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax6o 1758. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Axiomax-9o 2165 A variant of ax9 1942. 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 1943. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
 
Axiomax-10o 2166 Axiom ax-10o 2166 ("o" for "old") was the original version of ax-10 2167, before it was discovered (in May 2008) that the shorter ax-10 2167 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 1983. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
Axiomax-10 2167 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 2166 ("o" for "old") and was replaced with this shorter ax-10 2167 in May 2008. The old axiom is proved from this one as theorem ax10o 1983. Conversely, this axiom is proved from ax-10o 2166 as theorem ax10from10o 2204.

This axiom was proved redundant in July 2015. See theorem ax10 1976.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax10 1976. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Axiomax-11o 2168 Axiom ax-11o 2168 ("o" for "old") was the original version of ax-11 1753, before it was discovered (in Jan. 2007) that the shorter ax-11 1753 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 2168 (from which the ax-11 1753 instance follows by theorem ax11 2182.) The proof is by induction on formula length, using ax11eq 2220 and ax11el 2221 for the basis steps and ax11indn 2222, ax11indi 2223, and ax11inda 2227 for the induction steps. (This paragraph is true provided we use ax-10o 2166 in place of ax-10 2167.)

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 2020. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Axiomax-12o 2169 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 1956. (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 2170 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 1623; see theorem ax15 2047. Alternately, ax-17 1623 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1623. We retain ax-15 2170 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1623, 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 2047. (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 2171* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1623 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 4324), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1623; see theorem ax16 2071. Alternately, ax-17 1623 becomes logically redundant in the presence of this axiom, but without ax-17 1623 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 2171 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1623, 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 2071. (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: ax5 , ax6 , ax9from9o , ax11 , ax12from12o

Theorems ax11 2182 and ax12from12o 2183 require some intermediate theorems that are included in this section.

 
Theoremax4 2172 This theorem repeats sp 1755 under the name ax4 2172, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-4 2162. It is preferred that references to this theorem use the name sp 1755. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremax5 2173 Rederivation of axiom ax-5 1563 from ax-5o 2163 and other older axioms. See ax5o 1757 for the derivation of ax-5o 2163 from ax-5 1563. (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 2174 Rederivation of axiom ax-6 1736 from ax-6o 2164 and other older axioms. See ax6o 1758 for the derivation of ax-6o 2164 from ax-6 1736. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremax9from9o 2175 Rederivation of axiom ax-9 1661 from ax-9o 2165 and other older axioms. See ax9o 1943 for the derivation of ax-9o 2165 from ax-9 1661. 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 2176  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 2177 Inference version of ax-5o 2163. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theoremaecom-o 2178 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 1978 using ax-10o 2166. Unlike ax10from10o 2204, this version does not require ax-17 1623. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremaecoms-o 2179 A commutation rule for identical variable specifiers. Version of aecoms 1979 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 2180 All variables are effectively bound in an identical variable specifier. Version of hbae 1984 using ax-10o 2166. (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 2181 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 1991 using ax-10o 2166. (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 2182 Rederivation of axiom ax-11 1753 from ax-11o 2168, ax-10o 2166, and other older axioms. The proof does not require ax-16 2171 or ax-17 1623. See theorem ax11o 2020 for the derivation of ax-11o 2168 from ax-11 1753.

An open problem is whether we can prove this using ax-10 2167 instead of ax-10o 2166.

This proof uses newer axioms ax-5 1563 and ax-9 1661, 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 2163 and ax-9o 2165. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax12from12o 2183 Derive ax-12 1939 from ax-12o 2169 and other older axioms.

This proof uses newer axioms ax-5 1563 and ax-9 1661, 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 2163 and ax-9o 2165. (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 2184* 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 1623 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 1552, ax-5o 2163, ax-4 2162, ax-7 1741, ax-6o 2164, ax-8 1682, ax-12o 2169, ax-9o 2165, ax-10o 2166, ax-13 1719, ax-14 1721, ax-15 2170, ax-11o 2168, and ax-16 2171: in that system, we can derive any instance of ax-17 1623 not containing wff variables by induction on formula length, using ax17eq 2210 and ax17el 2216 for the basis together hbn 1784, hbal 1743, and hbim 1826. 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 2185 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 1623; see the proof of equid 1683. See equid1ALT 2203 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  x  =  x
 
Theoremsps-o 2186 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theoremhbequid 2187 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 2165.) (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 2188 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 1563, ax-8 1682, ax-12o 2169, and ax-gen 1552. This shows that this can be proved without ax9 1942, even though the theorem equid 1683 cannot be. A shorter proof using ax9 1942 is obtainable from equid 1683 and hbth 1558.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1662, which is used for the derivation of ax12o 1956, unless we consider ax-12o 2169 the starting axiom rather than ax-12 1939. (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
 
Theoremax46 2189 Proof of a single axiom that can replace ax-4 2162 and ax-6o 2164. See ax46to4 2190 and ax46to6 2191 for the re-derivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A. x  -.  A. x ph  ->  A. x ph )  ->  ph )
 
Theoremax46to4 2190 Re-derivation of ax-4 2162 from ax46 2189. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremax46to6 2191 Re-derivation of ax-6o 2164 from ax46 2189. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax67 2192 Proof of a single axiom that can replace both ax-6o 2164 and ax-7 1741. See ax67to6 2194 and ax67to7 2195 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. y A. x ph 
 ->  A. y ph )
 
Theoremnfa1-o 2193  x is not free in  A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ x A. x ph
 
Theoremax67to6 2194 Re-derivation of ax-6o 2164 from ax67 2192. Note that ax-6o 2164 and ax-7 1741 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax67to7 2195 Re-derivation of ax-7 1741 from ax67 2192. Note that ax-6o 2164 and ax-7 1741 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax467 2196 Proof of a single axiom that can replace ax-4 2162, ax-6o 2164, and ax-7 1741 in a subsystem that includes these axioms plus ax-5o 2163 and ax-gen 1552 (and propositional calculus). See ax467to4 2197, ax467to6 2198, and ax467to7 2199 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2189. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A. x A. y  -.  A. x A. y ph  ->  A. x ph )  ->  ph )
 
Theoremax467to4 2197 Re-derivation of ax-4 2162 from ax467 2196. Only propositional calculus is used by the re-derivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremax467to6 2198 Re-derivation of ax-6o 2164 from ax467 2196. Note that ax-6o 2164 and ax-7 1741 are not used by the re-derivation. The use of alimi 1565 (which uses ax-4 2162) is allowed since we have already proved ax467to4 2197. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax467to7 2199 Re-derivation of ax-7 1741 from ax467 2196. Note that ax-6o 2164 and ax-7 1741 are not used by the re-derivation. The use of alimi 1565 (which uses ax-4 2162) is allowed since we have already proved ax467to4 2197. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremequidqe 2200 equid 1683 with existential quantifier without using ax-4 2162 or ax-17 1623. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)
 |- 
 -.  A. y  -.  x  =  x
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