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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem19.42vv 1801* Version of 19.42 2000 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <->  ( ph  /\  E. x E. y ps )
 )
 
Theorem19.42vvv 1802* Version of 19.42 2000 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps )  <->  ( ph  /\  E. x E. y E. z ps ) )
 
Theoremexdistr2 1803* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps )  <->  E. x ( ph  /\ 
 E. y E. z ps ) )
 
Theorem3exdistr 1804* Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps  /\  ch )  <->  E. x ( ph  /\  E. y ( ps  /\  E. z ch ) ) )
 
Theorem4exdistr 1805* Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
 |-  ( E. x E. y E. z E. w ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  E. x ( ph  /\ 
 E. y ( ps 
 /\  E. z ( ch 
 /\  E. w th )
 ) ) )
 
Theoremspimeh 1806* Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
Theoremspimw 1807* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspimvw 1808* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  ps )
 
Theoremspnfw 1809 Weak version of sp 1883. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)
 |-  ( -.  ph  ->  A. x  -.  ph )   =>    |-  ( A. x ph  ->  ph )
 
Theoremspfalw 1810 Version of sp 1883 when  ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
 |- 
 -.  ph   =>    |-  ( A. x ph  -> 
 ph )
 
Theoremequs4v 1811* Version of equs4 2061 with a dv condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph ) )
 
Theoremcbvaliw 1812* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  A. y ps )
 
Theoremcbvalivw 1813* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  A. y ps )
 
1.4.8  Axiom scheme ax-7 (Equality)
 
Axiomax-7 1814 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1820). This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1527 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

Note that this axiom is still valid even when any two or all three of  x,  y, and  z are replaced with the same variable since they do not have any distinct variable (Metamath's $d) restrictions. Because of this, we say that these three variables are "bundled" (a term coined by Raph Levien). (Contributed by NM, 10-Jan-1993.)

 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Theoremequid 1815 Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
 |-  x  =  x
 
Theoremnfequid 1816 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. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
 |- 
 F/ y  x  =  x
 
Theoremequcomi 1817 Commutative law for equality. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  y  =  x )
 
Theoremequcom 1818 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
 |-  ( x  =  y  <-> 
 y  =  x )
 
Theoremequcoms 1819 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.)
 |-  ( x  =  y 
 ->  ph )   =>    |-  ( y  =  x 
 ->  ph )
 
Theoremequtr 1820 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( y  =  z 
 ->  x  =  z
 ) )
 
Theoremequtrr 1821 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  =  x 
 ->  z  =  y
 ) )
 
Theoremequequ1 1822 An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
 |-  ( x  =  y 
 ->  ( x  =  z  <-> 
 y  =  z ) )
 
Theoremequequ2 1823 An equivalence law for equality. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)
 |-  ( x  =  y 
 ->  ( z  =  x  <-> 
 z  =  y ) )
 
Theoremstdpc6 1824 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1825.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
 |- 
 A. x  x  =  x
 
Theoremstdpc7 1825 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1824.) Translated to traditional notation, it can be read: " x  =  y  ->  ( ph (
x ,  x )  ->  ph ( x ,  y ) ), provided that  y is free for  x in  ph ( x ,  x
)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
 |-  ( x  =  y 
 ->  ( [ x  /  y ] ph  ->  ph )
 )
 
Theoremequtr2 1826 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
 
Theoremequviniv 1827* A specialized version of equvini 2113 with a distinct variable restriction. (Contributed by Wolf Lammen, 8-Sep-2018.)
 |-  ( x  =  y 
 ->  E. z ( x  =  z  /\  y  =  z ) )
 
Theoremequvin 1828* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1861, ax-13 2026. (Revised by Wolf Lammen, 10-Jun-2019.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremax13b 1829 Two equivalent ways of expressing ax-13 2026. See the comment for ax-13 2026. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.)
 |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 ) 
 <->  ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) ) )
 
Theoremspfw 1830* Weak version of sp 1883. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  -> 
 ph )
 
Theoremspw 1831* Weak version of the specialization scheme sp 1883. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1883 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1883 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1855 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1883 are spfw 1830 (minimal distinct variable requirements), spnfw 1809 (when  x is not free in  -.  ph), spvw 1780 (when  x does not appear in  ph), sptruw 1651 (when  ph is true), and spfalw 1810 (when  ph is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
Theoremcbvalw 1832* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps 
 ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvalvw 1833* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvexvw 1834* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremalcomiw 1835* Weak version of alcom 1869. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremhbn1fw 1836* Weak version of ax-10 1861 from which we can prove any ax-10 1861 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps 
 ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( -.  A. y ps  ->  A. x  -.  A. y ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbn1w 1837* Weak version of hbn1 1862. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph  ->  A. x  -.  A. x ph )
 
Theoremhba1w 1838* Weak version of hba1 1924. See comments for ax10w 1849. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  A. x A. x ph )
 
Theoremhbe1w 1839* Weak version of hbe1 1863. See comments for ax10w 1849. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ph 
 ->  A. x E. x ph )
 
Theoremhbalw 1840* Weak version of hbal 1868. Uses only Tarski's FOL axiom schemes. Unlike hbal 1868, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  z 
 ->  ( ph  <->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremcbvaev 1841* Change bound variable in an equality with a dv condition. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  w )
 
1.4.9  Membership predicate
 
Syntaxwcel 1842 Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.

(The purpose of introducing 
wff  A  e.  B here is to allow us to express i.e. "prove" the wel 1843 of predicate calculus in terms of the wcel 1842 of set theory, so that we don't "overload" the  e. connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables  A and  B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2388 for more information on the set theory usage of wcel 1842.)

 wff  A  e.  B
 
Theoremwel 1843 Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read " x is an element of  y," " x is a member of  y," " x belongs to  y," or " y contains  x." Note: The phrase " y includes  x " means " x is a subset of  y;" to use it also for  x  e.  y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactical construction introduces a binary non-logical predicate symbol  e. (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for  e. apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

(Instead of introducing wel 1843 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1842. This lets us avoid overloading the  e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1843 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1842. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

 wff  x  e.  y
 
1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)
 
Axiomax-8 1844 Axiom of Left Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of an arbitrary binary predicate 
e., which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 30-Jun-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  ->  y  e.  z ) )
 
Theoremelequ1 1845 An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  <->  y  e.  z ) )
 
1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)
 
Axiomax-9 1846 Axiom of Right Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of an arbitrary binary predicate 
e., which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). (Contributed by NM, 21-Jun-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  ->  z  e.  y ) )
 
Theoremelequ2 1847 An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  <->  z  e.  y ) )
 
1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13

The original axiom schemes of Tarski's predicate calculus are ax-4 1652, ax-5 1725, ax6v 1772, ax-7 1814, ax-8 1844, and ax-9 1846, together with rule ax-gen 1639. See http://us.metamath.org/mpeuni/mmset.html#compare 1639. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85.

The axiom system of set.mm includes the auxiliary axiom schemes ax-10 1861, ax-11 1866, ax-12 1878, and ax-13 2026, which are not part of Tarski's axiom schemes. Each object language instance of them is provable from Tarski's axioms, so they are logically redundant. However, they are conjectured not to be provable directly as schemes from Tarski's axiom schemes using only Metamath's direct substitution rule. They are used to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and set metavariables, bundled or not, whose object-language instances are valid. (ax-12 1878 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.)

(There are additional predicate calculus axiom schemes included in set.mm such as ax-c5 31907, but they can all be proved as theorems from the above.)

Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the  x and  y in ax-6 1771 are bundled, but they are not in ax6v 1772. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1772 is the principal instance of ax-6 1771. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance  -.  A. x -.  x  =  x of ax-6 1771 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with. There is also a small economy in being able to state principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them).

Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-10 1861, ax-11 1866, ax-12 1878, and ax-13 2026. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 1861, ax-11 1866, ax-12 1878, or ax-13 2026 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.)

It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes.

The theorem schemes ax10w 1849, ax11w 1850, ax12w 1853, and ax13w 1856 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 1861, ax-11 1866, ax-12 1878, and ax-13 2026 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 1849, ax11w 1850, and ax12w 1853 is of the form  ( x  =  y  ->  ( ph  <->  ps ) ) where  ps is an auxiliary or "dummy" wff metavariable in which  x doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting conditions (1) and (2). The example ax12wdemo 1855 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this.

We also show the degenerate instances for axioms with bundled variables in ax11dgen 1851, ax12dgen 1854, ax13dgen1 1857, ax13dgen2 1858, ax13dgen3 1859, and ax13dgen4 1860. (Their proofs are trivial, but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-10 1861, ax-11 1866, ax-12 1878, and ax-13 2026 are schemes of Tarski's system, meaning that all object language instances they generate are theorems of Tarski's system.

It is interesting that Tarski used the bundled scheme ax-6 1771 in an older system, so it seems the main purpose of his later ax6v 1772 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-6 1771 as our official axiom, we show that the degenerate instance holds in ax6dgen 1848.

The case of sp 1883 is curious: originally an axiom of Tarski's system, it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the scheme form  A. x ph  ->  ph apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 1831, again requiring substitution instances of  ph that meet conditions (1) and (2) above. Note that our direct proof sp 1883 requires ax-12 1878, which is not part of Tarski's system.

 
Theoremax6dgen 1848 Tarski's system uses the weaker ax6v 1772 instead of the bundled ax-6 1771, so here we show that the degenerate case of ax-6 1771 can be derived. (Contributed by NM, 23-Apr-2017.)
 |- 
 -.  A. x  -.  x  =  x
 
Theoremax10w 1849* Weak version of ax-10 1861 from which we can prove any ax-10 1861 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph  ->  A. x  -.  A. x ph )
 
Theoremax11w 1850* Weak version of ax-11 1866 from which we can prove any ax-11 1866 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 1866, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax11dgen 1851 Degenerate instance of ax-11 1866 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( A. x A. x ph  ->  A. x A. x ph )
 
Theoremax12wlem 1852* Lemma for weak version of ax-12 1878. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 1853. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax12w 1853* Weak version of ax-12 1878 from which we can prove any ax-12 1878 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that  x and  y be distinct (unless  x does not occur in  ph). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for  ph, see ax12wdemo 1855. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 y  =  z  ->  ( ph  <->  ch ) )   =>    |-  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax12dgen 1854 Degenerate instance of ax-12 1878 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( x  =  x 
 ->  ( A. x ph  ->  A. x ( x  =  x  ->  ph )
 ) )
 
Theoremax12wdemo 1855* Example of an application of ax12w 1853 that results in an instance of ax-12 1878 for a contrived formula with mixed free and bound variables,  ( x  e.  y  /\  A. x
z  e.  x  /\  A. y A. z y  e.  x ), in place of  ph. The proof illustrates bound variable renaming with cbvalvw 1833 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)
 |-  ( x  =  y 
 ->  ( A. y ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )  ->  A. x ( x  =  y  ->  ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )
 ) ) )
 
Theoremax13w 1856* Weak version (principal instance) of ax-13 2026. (Because  y and  z don't need to be distinct, this actually bundles the principal instance and the degenerate instance  ( -.  x  =  y  ->  ( y  =  y  ->  A. x
y  =  y ) ).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 1849, ax11w 1850, and ax12w 1853. (Contributed by NM, 10-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax13dgen1 1857 Degenerate instance of ax-13 2026 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  z  ->  A. x  x  =  z )
 )
 
Theoremax13dgen2 1858 Degenerate instance of ax-13 2026 where bundled variables  x and  z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  x  ->  A. x  y  =  x )
 )
 
Theoremax13dgen3 1859 Degenerate instance of ax-13 2026 where bundled variables  y and  z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  y  ->  A. x  y  =  y )
 )
 
Theoremax13dgen4 1860 Degenerate instance of ax-13 2026 where bundled variables  x,  y, and  z have a common substitution. Uses only Tarski's FOL axiom schemes . (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
 )
 
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)

In this section we introduce four additional schemes ax-10 1861, ax-11 1866, ax-12 1878, and ax-13 2026 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "metalogical completeness," which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs and set metavariables ranging over object-language individual variables.

To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax10w 1849, ax11w 1850, ax12w 1853, and ax13w 1856, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all set metavariables to be mutually distinct) can be proved using only the schemes in Tarski's system S2.

An open problem is to show that these four additional schemes are mutually metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 1878 from all others has been shown, and independence of Tarski's ax-6 1771 from all others has been shown; see items 9a and 11 on http://us.metamath.org/award2003.html.

 
1.5.1  Axiom scheme ax-10 (Quantified Negation)
 
Axiomax-10 1861 Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 1849) but is used as an auxiliary axiom to achieve metalogical completeness. It means that  x is not free in  -.  A. x ph. (Contributed by NM, 21-May-2008.) Use its alias hbn1 1862 instead. (New usage is discouraged.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbn1 1862 Alias for ax-10 1861 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbe1 1863  x is not free in  E. x ph. (Contributed by NM, 24-Jan-1993.)
 |-  ( E. x ph  ->  A. x E. x ph )
 
Theoremnfe1 1864  x is not free in  E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x E. x ph
 
Theoremmodal-5 1865 The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
 |-  ( -.  A. x  -.  ph  ->  A. x  -.  A. x  -.  ph )
 
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)
 
Axiomax-11 1866 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 1850) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremalcoms 1867 Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
Theoremhbal 1868 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by NM, 12-Mar-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremalcom 1869 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theoremalrot3 1870 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y A. z ph  <->  A. y A. z A. x ph )
 
Theoremalrot4 1871 Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
 |-  ( A. x A. y A. z A. w ph  <->  A. z A. w A. x A. y ph )
 
Theoremhbald 1872 Deduction form of bound-variable hypothesis builder hbal 1868. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( A. y ps  ->  A. x A. y ps ) )
 
Theoremexcom 1873 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-12 1878, ax-10 1861, ax-6 1771, ax-7 1814 and ax-5 1725. (Revised by Wolf Lammen, 8-Jan-2018.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theoremexcomim 1874 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-12 1878, ax-10 1861, ax-6 1771, ax-7 1814 and ax-5 1725. (Revised by Wolf Lammen, 8-Jan-2018.)
 |-  ( E. x E. y ph  ->  E. y E. x ph )
 
Theoremexcom13 1875 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. z E. y E. x ph )
 
Theoremexrot3 1876 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. y E. z E. x ph )
 
Theoremexrot4 1877 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z E. w ph  <->  E. z E. w E. x E. y ph )
 
1.5.3  Axiom scheme ax-12 (Substitution)
 
Axiomax-12 1878 Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent  A. x ( x  =  y  ->  ph ) is a way of expressing " y substituted for  x in wff  ph " (cf. sb6 2197). 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.

The original version of this axiom was ax-c15 31913 and was replaced with this shorter ax-12 1878 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2111. Conversely, this axiom is proved from ax-c15 31913 as theorem ax12 31927.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 31913) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 1879 and ax12v2 2109 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 1853) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.)

 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax12v 1879* This is a version of ax-12 1878 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax12v2 2109 for the rederivation of ax-c15 31913 from this theorem. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1861 and ax-13 2026. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremax12vOLD 1880* Obsolete proof of ax12v 1879 as of 8-Dec-2019. (Contributed by Jim Kingdon, 15-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theorem19.8a 1881 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 1777 for a version requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 1883. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
 |-  ( ph  ->  E. x ph )
 
Theorem19.8aOLD 1882 Obsolete proof of 19.8a 1881 as of 8-Dec-2019. (Contributed by NM, 9-Jan-1993.) (Revised by Wolf Lammen, 13-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ph )
 
Theoremsp 1883 Specialization. A universally quantified wff implies the wff without a quantifier Axiom scheme B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Also appears as Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint).

For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2118.

This theorem shows that our obsolete axiom ax-c5 31907 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114.

It appears that this scheme cannot be derived directly from Tarski's axioms without auxiliary axiom scheme ax-12 1878. It is thought the best we can do using only Tarski's axioms is spw 1831. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)

 |-  ( A. x ph  -> 
 ph )
 
Theoremaxc4 1884 Show that the original axiom ax-c4 31908 can be derived from ax-4 1652 and others. See ax4 31918 for the rederivation of ax-4 1652 from ax-c4 31908.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

 |-  ( A. x (
 A. x ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremaxc7 1885 Show that the original axiom ax-c7 31909 can be derived from ax-10 1861 and others. See ax10 31919 for the rederivation of ax-10 1861 from ax-c7 31909.

Normally, axc7 1885 should be used rather than ax-c7 31909, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremaxc7e 1886 Abbreviated version of axc7 1885. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. x ph  ->  ph )
 
Theoremmodal-b 1887 The analog in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
 |-  ( ph  ->  A. x  -.  A. x  -.  ph )
 
Theoremspi 1888 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
 |- 
 A. x ph   =>    |-  ph
 
Theoremsps 1889 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theorem2sp 1890 A double specialization (see sp 1883). Another double specialization, closer to PM*11.1, is 2stdpc4 2119. (Contributed by BJ, 15-Sep-2018.)
 |-  ( A. x A. y ph  ->  ph )
 
Theoremspsd 1891 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theorem19.2g 1892 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1775 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
 |-  ( A. x ph  ->  E. y ph )
 
Theorem19.21bi 1893 Inference form of 19.21 1933 and also deduction form of sp 1883. (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.21bbi 1894 Inference removing double quantifier. Version of 19.21bi 1893 with two quanditiers. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  A. x A. y ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.23bi 1895 Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 1938. (Contributed by NM, 12-Mar-1993.)
 |-  ( E. x ph  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremnexr 1896 Inference form of 19.8a 1881. (Contributed by Jeff Hankins, 26-Jul-2009.)
 |- 
 -.  E. x ph   =>    |- 
 -.  ph
 
Theoremnfr 1897 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
 |-  ( F/ x ph  ->  ( ph  ->  A. x ph ) )
 
Theoremnfri 1898 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfrd 1899 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremalimd 1900 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1653. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38860
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