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Theorem List for Metamath Proof Explorer - 1701-1800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem19.8wOLD 1701 Obsolete version of 19.8w 1668 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x ph )
 
Theoremspw 1702* Weak version of specialization scheme sp 1759. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1759 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1759 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1734 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1759 are spfw 1699 (minimal distinct variable requirements), spnfw 1678 (when  x is not free in  -.  ph), spvw 1674 (when  x does not appear in  ph), sptruw 1679 (when  ph is true), and spfalw 1680 (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 )
 
TheoremspwOLD 1703* Obsolete proof of spw 1702 as of 27-Feb-2018. (Contributed by NM, 9-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
TheoremspvwOLD 1704* Obsolete version of spvw 1674 as of 4-Dec-2017. (Contributed by NM, 10-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theorem19.3vOLD 1705* Obsolete version of 19.3v 1673 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x ph  <->  ph )
 
Theorem19.9vOLD 1706* Obsolete version of 19.9v 1672 as of 4-Dec-2017. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E. x ph  <->  ph )
 
TheoremexlimivOLD 1707* Obsolete version of exlimiv 1641 as of 4-Dec-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
TheoremspfalwOLD 1708 Obsolete proof of spfalw 1680 as of 25-Dec-2017. (Contributed by NM, 23-Apr-1017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 -.  ph   =>    |-  ( A. x ph  -> 
 ph )
 
Theorem19.2OLD 1709 Obsolete version of 19.2 1667 as of 4-Dec-2017. (Contributed by NM, 2-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  E. x ph )
 
Theoremcbvalw 1710* 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 1711* 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 )
 
TheoremcbvalvwOLD 1712* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvexvw 1713* 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 1714* Weak version of alcom 1748. 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 1715* Weak version of ax-6 1740 from which we can prove any ax-6 1740 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 )
 
Theoremhbn1fwOLD 1716* Obsolete proof of hbn1fw 1715 as of 28-Feb-2018. (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 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 1717* Weak version of hbn1 1741. 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 1718* Weak version of hba1 1800. See comments for ax6w 1728. 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 1719* Weak version of hbe1 1742. See comments for ax6w 1728. 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 1720* Weak version of hbal 1747. Uses only Tarski's FOL axiom schemes. Unlike hbal 1747, 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 )
 
1.4.8  Membership predicate
 
Syntaxwcel 1721 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 1722 of predicate calculus in terms of the wceq 1649 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 2391 for more information on the set theory usage of wcel 1721.)

 wff  A  e.  B
 
Theoremwel 1722 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 1722 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 1721. This lets us avoid overloading the  e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1722 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1721. 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
 
1.4.9  Axiom scheme ax-13 (Left Equality for Binary Predicate)
 
Axiomax-13 1723 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, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  ->  y  e.  z ) )
 
Theoremelequ1 1724 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  <->  y  e.  z ) )
 
1.4.10  Axiom scheme ax-14 (Right Equality for Binary Predicate)
 
Axiomax-14 1725 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, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  ->  z  e.  y ) )
 
Theoremelequ2 1726 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  <->  z  e.  y ) )
 
1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12

The orginal axiom schemes of Tarski's predicate calculus are ax-5 1563, ax-17 1623, ax9v 1663, ax-8 1683, ax-13 1723, and ax-14 1725, together with rule ax-gen 1552. See http://us.metamath.org/mpeuni/mmset.html#compare 1552. 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-6 1740, ax-7 1745, ax-12 1946, and ax-11 1757, which are not part of Tarski's axiom schemes. They are used (and we conjecture are required) 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-11 1757 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-4 2185, 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 ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the  x and  y in ax-9 1662 are bundled, but they are not in ax9v 1663. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax9v 1663 is the principal instance of ax-9 1662. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance  -.  A. x -.  x  =  x of ax-9 1662 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($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-6 1740, ax-7 1745, ax-11 1757, and ax-12 1946 . "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-6 1740, ax-7 1745, ax-11 1757, or ax-12 1946 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 new theorem schemes ax6w 1728, ax7w 1729, ax11w 1732, and ax12w 1735 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-6 1740, ax-7 1745, ax-11 1757, and ax-12 1946 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax6w 1728, ax7w 1729, and ax11w 1732 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 ax11wdemo 1734 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 ax7dgen 1730, ax11dgen 1733, ax12dgen1 1736, ax12dgen2 1737, ax12dgen3 1738, and ax12dgen4 1739. (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-6 1740, ax-7 1745, ax-11 1757, and ax-12 1946 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-9 1662 in an older system, so it seems the main purpose of his later ax9v 1663 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-9 1662 as our official axiom, we show that the degenerate instance holds in ax9dgen 1727.

The case of sp 1759 is curious: originally an axiom of Tarski's system, it was proved redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the compact scheme form  A. x ph  ->  ph apparently cannot be proved directly from Tarski's other axioms. The best we can do seems to be spw 1702, again requiring substitution instances of  ph that meet conditions (1) and (2) above. Note that our direct proof sp 1759 requires ax-11 1757, which is not part of Tarski's system.

 
Theoremax9dgen 1727 Tarski's system uses the weaker ax9v 1663 instead of the bundled ax-9 1662, so here we show that the degenerate case of ax-9 1662 can be derived. (Contributed by NM, 23-Apr-2017.)
 |- 
 -.  A. x  -.  x  =  x
 
Theoremax6w 1728* Weak version of ax-6 1740 from which we can prove any ax-6 1740 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 )
 
Theoremax7w 1729* Weak version of ax-7 1745 from which we can prove any ax-7 1745 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1745, 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 )
 
Theoremax7dgen 1730 Degenerate instance of ax-7 1745 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 )
 
Theoremax11wlem 1731* Lemma for weak version of ax-11 1757. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax11w 1732. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax11w 1732* Weak version of ax-11 1757 from which we can prove any ax-11 1757 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 ax11wdemo 1734. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 y  =  z  ->  ( ph  <->  ch ) )   =>    |-  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax11dgen 1733 Degenerate instance of ax-11 1757 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 )
 ) )
 
Theoremax11wdemo 1734* Example of an application of ax11w 1732 that results in an instance of ax-11 1757 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 1711 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 )
 ) ) )
 
Theoremax12w 1735* Weak version (principal instance) of ax-12 1946. (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 ax6w 1728, ax7w 1729, and ax11w 1732. (Contributed by NM, 10-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax12dgen1 1736 Degenerate instance of ax-12 1946 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 )
 )
 
Theoremax12dgen2 1737 Degenerate instance of ax-12 1946 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 )
 )
 
Theoremax12dgen3 1738 Degenerate instance of ax-12 1946 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 )
 )
 
Theoremax12dgen4 1739 Degenerate instance of ax-12 1946 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-6 1740, ax-7 1745, ax-11 1757, and ax-12 1946 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 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 ax6w 1728, ax7w 1729, ax12w 1735, and ax11w 1732, 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-11 1757 from all others has been shown, and independence of Tarski's ax-9 1662 from all others has been shown; see items 9a and 11 on http://us.metamath.org/award2003.html.

 
1.5.1  Axiom scheme ax-6 (Quantified Negation)
 
Axiomax-6 1740 Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax6w 1728) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbn1 1741  x is not free in  -.  A. x ph. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbe1 1742  x is not free in  E. x ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ph  ->  A. x E. x ph )
 
Theoremnfe1 1743  x is not free in  E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x E. x ph
 
Theoremmodal-5 1744 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-7 (Quantifier Commutation)
 
Axiomax-7 1745 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 ax7w 1729) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theorema7s 1746 Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
Theoremhbal 1747 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremalcom 1748 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theoremalrot3 1749 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 1750 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 1751 Deduction form of bound-variable hypothesis builder hbal 1747. (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 1752 Theorem 19.11 of [Margaris] p. 89. Revised to remove dependency on ax-11 1757, ax-6 1740, ax-9 1662, ax-8 1683 and ax-17 1623. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 8-Jan-2018.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theoremexcomim 1753 One direction of Theorem 19.11 of [Margaris] p. 89. Revised to remove dependency on ax-11 1757, ax-6 1740, ax-9 1662, ax-8 1683 and ax-17 1623. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Revised by Wolf Lammen, 8-Jan-2018.)
 |-  ( E. x E. y ph  ->  E. y E. x ph )
 
Theoremexcom13 1754 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 1755 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. y E. z E. x ph )
 
Theoremexrot4 1756 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-11 (Substitution)
 
Axiomax-11 1757 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 2148). 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-11o 2191 ("o" for "old") and was replaced with this shorter ax-11 1757 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 2047. Conversely, this axiom is proved from ax-11o 2191 as theorem ax11 2205.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2191) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax11v 2145 and ax11v2 2045 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax11w 1732) 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 )
 ) )
 
Theorem19.8a 1758 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 13-Jan-2018.)
 |-  ( ph  ->  E. x ph )
 
Theoremsp 1759 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 2073.

This theorem shows that our obsolete axiom ax-4 2185 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 auxilliary axiom scheme ax-11 1757. It is thought the best we can do using only Tarski's axioms is spw 1702. (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 )
 
TheoremspOLD 1760 Obsolete proof of sp 1759 as of 23-Dec-2017. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremax5o 1761 Show that the original axiom ax-5o 2186 can be derived from ax-5 1563 and others. See ax5 2196 for the rederivation of ax-5 1563 from ax-5o 2186.

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 ) )
 
Theoremax6o 1762 Show that the original axiom ax-6o 2187 can be derived from ax-6 1740 and others. See ax6 2197 for the rederivation of ax-6 1740 from ax-6o 2187.

Normally, ax6o 1762 should be used rather than ax-6o 2187, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theorema6e 1763 Abbreviated version of ax6o 1762. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. x ph  ->  ph )
 
Theoremmodal-b 1764 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 1765 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
 |- 
 A. x ph   =>    |-  ph
 
Theoremsps 1766 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theoremspsd 1767 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theorem19.8aOLD 1768 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  E. x ph )
 
Theorem19.2g 1769 Theorem 19.2 of [Margaris] p. 89, generalized to use two set variables. (Contributed by O'Cat, 31-Mar-2008.)
 |-  ( A. x ph  ->  E. y ph )
 
Theorem19.21bi 1770 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.23bi 1771 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ph  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremnexr 1772 Inference from 19.8a 1758. (Contributed by Jeff Hankins, 26-Jul-2009.)
 |- 
 -.  E. x ph   =>    |- 
 -.  ph
 
Theoremnfr 1773 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
 |-  ( F/ x ph  ->  ( ph  ->  A. x ph ) )
 
Theoremnfri 1774 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfrd 1775 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 1776 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
Theoremalrimi 1777 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremnfd 1778 Deduce that  x is not free in  ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  F/ x ps )
 
Theoremnfdh 1779 Deduce that  x is not free in  ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  F/ x ps )
 
Theoremalrimdd 1780 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremalrimd 1781 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremeximd 1782 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theoremnexd 1783 Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremalbid 1784 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbid 1785 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremnfbidf 1786 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( F/ x ps  <->  F/ x ch )
 )
 
Theorem19.3 1787 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   =>    |-  ( A. x ph  <->  ph )
 
Theorem19.9ht 1788 A closed version of 19.9 1793. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( E. x ph  -> 
 ph ) )
 
Theorem19.9t 1789 A closed version of 19.9 1793. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)
 |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
 
Theorem19.9h 1790 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. x ph  <->  ph )
 
Theorem19.9hOLD 1791 Obsolete proof of 19.9h 1790 as of 5-Jan-2018. (Contributed by FL, 24-Mar-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. x ph  <->  ph )
 
Theorem19.9d 1792 A deduction version of one direction of 19.9 1793. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( ps  ->  F/ x ph )   =>    |-  ( ps  ->  ( E. x ph  ->  ph )
 )
 
Theorem19.9 1793 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |- 
 F/ x ph   =>    |-  ( E. x ph  <->  ph )
 
Theorem19.9OLD 1794 Obsolete proof of 19.9 1793 as of 30-Dec-2017. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 F/ x ph   =>    |-  ( E. x ph  <->  ph )
 
Theoremhbnt 1795 Closed theorem version of bound-variable hypothesis builder hbn 1797. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( -.  ph  ->  A. x  -.  ph )
 )
 
TheoremhbntOLD 1796 Obsolete proof of hbnt 1795 as of 3-Mar-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( -.  ph  ->  A. x  -.  ph )
 )
 
Theoremhbn 1797 If  x is not free in  ph, it is not free in  -.  ph. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( -.  ph  ->  A. x  -.  ph )
 
TheoremhbnOLD 1798 Obsolete proof of hbn 1797 as of 16-Dec-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( -.  ph  ->  A. x  -.  ph )
 
Theorem19.9htOLD 1799 Obsolete proof of 19.9ht 1788 as of 3-Mar-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( E. x ph  -> 
 ph ) )
 
Theoremhba1 1800  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.) (Proof shortened by Wolf Lammen, 15-Dec-2017.)
 |-  ( A. x ph  ->  A. x A. x ph )
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