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Theorem List for Metamath Proof Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmerco1lem14 1501 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( (
 ph  ->  ps )  ->  ps )  ->  ch )  ->  ( ph  ->  ch ) )
 
Theoremmerco1lem15 1502 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ( ch  ->  ps )
 ) )
 
Theoremmerco1lem16 1503 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ( ps  ->  ch )
 )  ->  ta )  ->  ( ( ph  ->  ch )  ->  ta )
 )
 
Theoremmerco1lem17 1504 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ( ( ph  ->  ps )  -> 
 ph )  ->  ch )  ->  ta )  ->  (
 ( ph  ->  ch )  ->  ta ) )
 
Theoremmerco1lem18 1505 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ( ps  ->  ph )  ->  ( ps  ->  ch ) ) )
 
Theoremretbwax1 1506 tbw-ax1 1471 rederived from merco1 1484.

This theorem, along with retbwax2 1487, retbwax3 1494, and retbwax4 1486, shows that merco1 1484 with ax-mp 8 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom
 
Theoremmerco2 1507 A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1484. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( ( ( ph  ->  ps )  ->  (
 (  F.  ->  ch )  ->  th ) )  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( et  ->  ph )
 ) ) )
 
Theoremmercolem1 1508 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( ps  ->  ( th  ->  ch ) ) )
 
Theoremmercolem2 1509 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ph )  ->  ( ch  ->  ( th  ->  ph ) ) )
 
Theoremmercolem3 1510 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ps  ->  ch )  ->  ( ps  ->  ( ph  ->  ch )
 ) )
 
Theoremmercolem4 1511 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( th  ->  ( et  ->  ph ) ) 
 ->  ( ( ( th  ->  ch )  ->  ph )  ->  ( ta  ->  ( et  ->  ph ) ) ) )
 
Theoremmercolem5 1512 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( th  ->  (
 ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) )
 
Theoremmercolem6 1513 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) ) 
 ->  ( ps  ->  ( ph  ->  ch ) ) )
 
Theoremmercolem7 1514 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( (
 ( ph  ->  ch )  ->  ( th  ->  ps )
 )  ->  ( th  ->  ps ) ) )
 
Theoremmercolem8 1515 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ( ph  ->  ch ) )  ->  ( ta  ->  ( th  ->  (
 ph  ->  ch ) ) ) ) )
 
Theoremre1tbw1 1516 tbw-ax1 1471 rederived from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremre1tbw2 1517 tbw-ax2 1472 rederived from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremre1tbw3 1518 tbw-ax3 1473 rederived from merco2 1507. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ph )  -> 
 ph )
 
Theoremre1tbw4 1519 tbw-ax4 1474 rederived from merco2 1507.

This theorem, along with re1tbw1 1516, re1tbw2 1517, and re1tbw3 1518, shows that merco2 1507, along with ax-mp 8, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  (  F.  ->  ph )
 
1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms
 
Theoremrb-bijust 1520 Justification for rb-imdf 1521. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  <->  ps )  <->  -.  ( -.  ( -.  ph  \/  ps )  \/  -.  ( -.  ps  \/  ph ) ) )
 
Theoremrb-imdf 1521 The definition of implication, in terms of  \/ and  -.. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  ( -.  ( -.  ( ph  ->  ps )  \/  ( -.  ph  \/  ps ) )  \/  -.  ( -.  ( -.  ph  \/  ps )  \/  ( ph  ->  ps ) ) )
 
Theoremanmp 1522 Modus ponens for  \/  -. axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( -.  ph  \/  ps )   =>    |- 
 ps
 
Theoremrb-ax1 1523 The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ( -. 
 ps  \/  ch )  \/  ( -.  ( ph  \/  ps )  \/  ( ph  \/  ch ) ) )
 
Theoremrb-ax2 1524 The second of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ( ph  \/  ps )  \/  ( ps  \/  ph ) )
 
Theoremrb-ax3 1525 The third of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ph  \/  ( ps  \/  ph )
 )
 
Theoremrb-ax4 1526 The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ( ph  \/  ph )  \/  ph )
 
Theoremrbsyl 1527 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ps  \/  ch )   &    |-  ( ph  \/  ps )   =>    |-  ( ph  \/  ch )
 
Theoremrblem1 1528 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ph  \/  ps )   &    |-  ( -.  ch  \/  th )   =>    |-  ( -.  ( ph  \/  ch )  \/  ( ps  \/  th ) )
 
Theoremrblem2 1529 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ( ch 
 \/  ph )  \/  ( ch  \/  ( ph  \/  ps ) ) )
 
Theoremrblem3 1530 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ( ch 
 \/  ph )  \/  (
 ( ch  \/  ps )  \/  ph ) )
 
Theoremrblem4 1531 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ph  \/  th )   &    |-  ( -.  ps  \/  ta )   &    |-  ( -.  ch  \/  et )   =>    |-  ( -.  ( (
 ph  \/  ps )  \/  ch )  \/  (
 ( et  \/  ta )  \/  th ) )
 
Theoremrblem5 1532 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  ( -. 
 -.  ph  \/  ps )  \/  ( -.  -.  ps  \/  ph ) )
 
Theoremrblem6 1533 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  ( -.  ( -.  ph  \/  ps )  \/  -.  ( -.  ps  \/  ph ) )   =>    |-  ( -.  ph  \/  ps )
 
Theoremrblem7 1534 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  ( -.  ( -.  ph  \/  ps )  \/  -.  ( -.  ps  \/  ph ) )   =>    |-  ( -.  ps  \/  ph )
 
Theoremre1axmp 1535 ax-mp 8 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  ps
 
Theoremre2luk1 1536 luk-1 1426 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremre2luk2 1537 luk-2 1427 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( -.  ph  -> 
 ph )  ->  ph )
 
Theoremre2luk3 1538 luk-3 1428 derived from Russell-Bernays'.

This theorem, along with re1axmp 1535, re2luk1 1536, and re2luk2 1537 shows that rb-ax1 1523, rb-ax2 1524, rb-ax3 1525, and rb-ax4 1526, along with anmp 1522, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( ph  ->  ( -.  ph  ->  ps )
 )
 
1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)

The Greek Stoics developed a system of logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic" https://www.historyoflogic.com/logic-stoics.htm). For more about Aristotle's system, see barbara 2351 and related theorems.

A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 8, modus tollendo tollens (modus tollens) mto 169, modus ponendo tollens I mpto1 1539, modus ponendo tollens II mpto2 1540, and modus tollendo ponens (exclusive-or version) mtp-xor 1542. The first is an axiom, the second is already proved; in this section we prove the other three. Since we assume or prove all of indemonstrables, the system of logic we use here is as at least as strong as the set of Stoic indemonstrables. Note that modus tollendo ponens mtp-xor 1542 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtp-or 1544. This set of indemonstrables is not the entire system of Stoic logic.

 
Theoremmpto1 1539 Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mpto2 1540) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.)
 |-  ph   &    |- 
 -.  ( ph  /\  ps )   =>    |- 
 -.  ps
 
Theoremmpto2 1540 Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or  \/_. See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.)
 |-  ph   &    |-  ( ph  \/_  ps )   =>    |- 
 -.  ps
 
Theoremmpto2OLD 1541 Obsolete version of mpto2 1540 as of 12-Nov-2017. (Contributed by David A. Wheeler, 3-Jul-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ph   &    |-  ( ph  \/_  ps )   =>    |- 
 -.  ps
 
Theoremmtp-xor 1542 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, one of the five "indemonstrables" in Stoic logic. The rule says, "if  ph is not true, and either  ph or  ps (exclusively) are true, then  ps must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtp-or 1544. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1540, that is, it is exclusive-or df-xor 1311), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mpto2 1540), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
 |- 
 -.  ph   &    |-  ( ph  \/_  ps )   =>    |- 
 ps
 
Theoremmtp-xorOLD 1543 Obsolete version of mtp-xor 1542 as of 11-Nov-2017. (Contributed by David A. Wheeler, 4-Jul-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 -.  ph   &    |-  ( ph  \/_  ps )   =>    |- 
 ps
 
Theoremmtp-or 1544 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1542, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if  ph is not true, and  ph or  ps (or both) are true, then  ps must be true." An alternative phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
 |- 
 -.  ph   &    |-  ( ph  \/  ps )   =>    |- 
 ps
 
Theoremmtp-orOLD 1545 Obsolete version of mtp-or 1544 as of 11-Nov-2017. (Contributed by David A. Wheeler, 3-Jul-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 -.  ph   &    |-  ( ph  \/  ps )   =>    |- 
 ps
 
1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)

Here we extend the language of wffs with predicate calculus, which allows us to talk about individual objects in a domain of discussion (which for us will be the universe of all sets, so we call them "set variables") and make true/false statements about predicates, which are relationships between objects, such as whether or not two objects are equal. In addition, we introduce universal quantification ("for all") in order to make statements about whether a wff holds for every object in the domain of discussion. Later we introduce existential quantification ("there exists", df-ex 1548) which is defined in terms of universal quantification.

Our axioms are really axiom schemes, and our wff and set variables are metavariables ranging over expressions in an underlying "object language." This is explained here: http://us.metamath.org/mpeuni/mmset.html#axiomnote

Our axiom system starts with the predicate calculus axiom schemes system S2 of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate Logic with Identity" [Tarski]. System S2 is defined in the last paragraph on p. 77, and repeated on p. 81 of [KalishMontague]. We do not include scheme B5 (our sp 1759) since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1702 below).

Theorem spw 1702 can be used to prove any instance of sp 1759 having no wff metavariables and mutually distinct set variables. However, it seems that sp 1759 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 1759 as part of what we call "Tarski's system" because we want it to be the smallest set of axioms that is logically complete with no redundancies. We later prove sp 1759 as theorem ax4 2195 using the auxiliary axioms that make our system metalogically complete.

Our version of Tarski's system S2 consists of propositional calculus plus ax-gen 1552, ax-5 1563, ax-17 1623, ax-9 1662, ax-8 1683, ax-13 1723, and ax-14 1725. The last 3 are equality axioms that represent 3 sub-schemes of Tarski's scheme B8. Due to its side-condition ("where  ph is an atomic formula and  ps is obtained by replacing an occurrence of the variable  x by the variable  y"), we cannot represent his B8 directly without greatly complicating our scheme language, but the simpler schemes ax-8 1683, ax-13 1723, and ax-14 1725 are sufficient for set theory.

Tarski's system is exactly equivalent to the traditional axiom system in most logic textbooks but has the advantage of being easy to manipulate with a computer program, and its simpler metalogic (with no built-in notions of free variable and proper substitution) is arguably easier for a non-logician human to follow step by step in a proof.

However, in our system that derives schemes (rather than object language theorems) from other schemes, Tarski's S2 is not complete. For example, we cannot derive scheme sp 1759, even though (using spw 1702) we can derive all instances of it that don't involve wff metavariables or bundled set metavariables. (Two set metavariables are "bundled" if they can be substituted with the same set metavariable i.e. do not have a $d distinct variable proviso.) Later we will introduce auxiliary axiom schemes ax-6 1740, ax-7 1745, ax-12 1946, and ax-11 1757 that are metatheorems of Tarski's system (i.e. are logically redundant) but which give our system the property of "metalogical completeness," allowing us to prove directly (instead of, say, by induction on formula length) all possible schemes that can be expressed in our language.

 
1.4.1  Universal quantifier; define "exists" and "not free"
 
Syntaxwal 1546 Extend wff definition to include the universal quantifier ('for all').  A. x ph is read " ph (phi) is true for all  x." Typically, in its final application  ph would be replaced with a wff containing a (free) occurrence of the variable  x, for example  x  =  y. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of  x. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
 wff  A. x ph
 
Syntaxwex 1547 Extend wff definition to include the existential quantifier ("there exists").
 wff  E. x ph
 
Definitiondf-ex 1548 Define existential quantification.  E. x ph means "there exists at least one set  x such that  ph is true." Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ph  <->  -.  A. x  -.  ph )
 
Theoremalnex 1549 Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  -.  ph  <->  -. 
 E. x ph )
 
Syntaxwnf 1550 Extend wff definition to include the not-free predicate.
 wff  F/ x ph
 
Definitiondf-nf 1551 Define the not-free predicate for wffs. This is read " x is not free in  ph". Not-free means that the value of  x cannot affect the value of  ph, e.g., any occurrence of  x in  ph is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 2075). An example of where this is used is stdpc5 1812. See nf2 1885 for an alternative definition which does not involve nested quantifiers on the same variable.

Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.

To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example,  x is effectively not free in the bare expression  x  =  x (see nfequid 1686), even though  x would be considered free in the usual textbook definition, because the value of  x in the expression  x  =  x cannot affect the truth of the expression (and thus substitution will not change the result).

This predicate only applies to wffs. See df-nfc 2529 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)

 |-  ( F/ x ph  <->  A. x ( ph  ->  A. x ph ) )
 
1.4.2  Rule scheme ax-gen (Generalization)
 
Axiomax-gen 1552 Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved  x  =  x, we can conclude  A. x x  =  x or even  A. y
x  =  x. Theorem allt 26055 shows the special case  A. x  T.. Theorem spi 1765 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 5-Aug-1993.)
 |-  ph   =>    |- 
 A. x ph
 
Theoremgen2 1553 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
 |-  ph   =>    |- 
 A. x A. y ph
 
Theoremmpg 1554 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
 |-  ( A. x ph  ->  ps )   &    |-  ph   =>    |- 
 ps
 
Theoremmpgbi 1555 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
 |-  ( A. x ph  <->  ps )   &    |-  ph   =>    |- 
 ps
 
Theoremmpgbir 1556 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
 |-  ( ph  <->  A. x ps )   &    |-  ps   =>    |-  ph
 
Theoremnfi 1557 Deduce that  x is not free in  ph from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  A. x ph )   =>    |- 
 F/ x ph
 
Theoremhbth 1558 No variable is (effectively) free in a theorem.

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form  |-  ( ph  ->  A. x ph ) from smaller formulas of this form. These are useful for constructing hypotheses that state " x is (effectively) not free in  ph." (Contributed by NM, 5-Aug-1993.)

 |-  ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfth 1559 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- 
 F/ x ph
 
Theoremnftru 1560 The true constant has no free variables. (This can also be proven in one step with nfv 1626, but this proof does not use ax-17 1623.) (Contributed by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x  T.
 
Theoremnex 1561 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
 |- 
 -.  ph   =>    |- 
 -.  E. x ph
 
Theoremnfnth 1562 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
 |- 
 -.  ph   =>    |- 
 F/ x ph
 
1.4.3  Axiom scheme ax-5 (Quantified Implication)
 
Axiomax-5 1563 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremalim 1564 Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremalimi 1565 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theorem2alimi 1566 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x A. y ph  ->  A. x A. y ps )
 
Theoremal2imi 1567 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x ph 
 ->  ( A. x ps  ->  A. x ch )
 )
 
Theoremalanimi 1568 Variant of al2imi 1567 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x ph 
 /\  A. x ps )  ->  A. x ch )
 
Theoremalimdh 1569 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
Theoremalbi 1570 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( A. x ph  <->  A. x ps )
 )
 
Theoremalrimih 1571 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremalbii 1572 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
 |-  ( ph  <->  ps )   =>    |-  ( A. x ph  <->  A. x ps )
 
Theorem2albii 1573 Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
 |-  ( ph  <->  ps )   =>    |-  ( A. x A. y ph  <->  A. x A. y ps )
 
Theoremhbxfrbi 1574 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2507 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfbii 1575 Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  <->  ps )   =>    |-  ( F/ x ph  <->  F/ x ps )
 
Theoremnfxfr 1576 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  <->  ps )   &    |-  F/ x ps   =>    |-  F/ x ph
 
Theoremnfxfrd 1577 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  ->  F/ x ps )   =>    |-  ( ch  ->  F/ x ph )
 
Theoremalex 1578 Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  <->  -.  E. x  -.  ph )
 
Theorem2nalexn 1579 Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  A. x A. y ph  <->  E. x E. y  -.  ph )
 
Theoremexnal 1580 Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x  -.  ph  <->  -. 
 A. x ph )
 
Theoremexim 1581 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
 
Theoremeximi 1582 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  E. x ps )
 
Theorem2eximi 1583 Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  E. x E. y ps )
 
Theoremeximii 1584 Inference associated with eximi 1582. (Contributed by BJ, 3-Feb-2018.)
 |- 
 E. x ph   &    |-  ( ph  ->  ps )   =>    |- 
 E. x ps
 
Theoremalinexa 1585 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
 |-  ( A. x (
 ph  ->  -.  ps )  <->  -. 
 E. x ( ph  /\ 
 ps ) )
 
Theoremalexn 1586 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
 |-  ( A. x E. y  -.  ph  <->  -.  E. x A. y ph )
 
Theorem2exnexn 1587 Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.)
 |-  ( E. x A. y ph  <->  -.  A. x E. y  -.  ph )
 
Theoremexbi 1588 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremexbii 1589 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x ph  <->  E. x ps )
 
Theorem2exbii 1590 Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x E. y ph  <->  E. x E. y ps )
 
Theorem3exbii 1591 Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x E. y E. z ph  <->  E. x E. y E. z ps )
 
Theoremexanali 1592 A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)
 |-  ( E. x (
 ph  /\  -.  ps )  <->  -. 
 A. x ( ph  ->  ps ) )
 
Theoremexancom 1593 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  E. x ( ps  /\  ph ) )
 
Theoremalrimdh 1594 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremeximdh 1595 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theoremnexdh 1596 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremalbidh 1597 Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidh 1598 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremexsimpl 1599 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( E. x (
 ph  /\  ps )  ->  E. x ph )
 
Theorem19.26 1600 Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  A. x ps ) )
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