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Theorem List for Metamath Proof Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremre1tbw2 1501 tbw-ax2 1456 rederived from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremre1tbw3 1502 tbw-ax3 1457 rederived from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ph )  -> 
 ph )
 
Theoremre1tbw4 1503 tbw-ax4 1458 rederived from merco2 1491.

This theorem, along with re1tbw1 1500, re1tbw2 1501, and re1tbw3 1502, shows that merco2 1491, 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 1504 Justification for rb-imdf 1505. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  <->  ps )  <->  -.  ( -.  ( -.  ph  \/  ps )  \/  -.  ( -.  ps  \/  ph ) ) )
 
Theoremrb-imdf 1505 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 1506 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 1507 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 1508 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 1509 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 1510 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 1511 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 1512 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 1513 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 1514 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 1515 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 1516 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 1517 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 1518 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 1519 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 1520 luk-1 1410 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 1521 luk-2 1411 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( -.  ph  -> 
 ph )  ->  ph )
 
Theoremre2luk3 1522 luk-3 1412 derived from Russell-Bernays'.

This theorem, along with re1axmp 1519, re2luk1 1520, and re2luk2 1521 shows that rb-ax1 1507, rb-ax2 1508, rb-ax3 1509, and rb-ax4 1510, along with anmp 1506, 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 2240 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 167, modus ponendo tollens I mpto1 1523, modus ponendo tollens II mpto2 1524, and modus tollendo ponens (exclusive-or version) mtp-xor 1525. 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 1525 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 1526. This set of indemonstrables is not the entire system of Stoic logic.

 
Theoremmpto1 1523 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 1524) 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 1524 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.)
 |-  ph   &    |-  ( ph \/_ ps )   =>    |- 
 -.  ps
 
Theoremmtp-xor 1525 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 1526. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1524, that is, it is exclusive-or df-xor 1296), 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 1524), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.)
 |- 
 -.  ph   &    |-  ( ph \/_ ps )   =>    |- 
 ps
 
Theoremmtp-or 1526 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1525, 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.)
 |- 
 -.  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 1529) 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 1716) since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1660 below).

Theorem spw 1660 can be used to prove any instance of sp 1716 having no wff metavariables and mutually distinct set variables. However, it seems that sp 1716 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 1716 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 1716 as theorem ax4 2084 using the auxiliary axioms that make our system metalogically complete.

Our version of Tarski's system S2 consists of propositional calculus plus ax-gen 1533, ax-5 1544, ax-17 1603, ax-9 1635, ax-8 1643, ax-13 1686, and ax-14 1688. 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 1643, ax-13 1686, and ax-14 1688 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 1716, even though (using spw 1660) 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 1703, ax-7 1708, ax-12 1866, and ax-11 1715 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 1527 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 1528 Extend wff definition to include the existential quantifier ("there exists").
 wff  E. x ph
 
Definitiondf-ex 1529 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 1530 Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  -.  ph  <->  -. 
 E. x ph )
 
Syntaxwnf 1531 Extend wff definition to include the not-free predicate.
 wff  F/ x ph
 
Definitiondf-nf 1532 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 1966). An example of where this is used is stdpc5 1793. See nf2 1798 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 1645), 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 2408 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 1533 Rule of Generalization. The postulated inference rule of pure 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 24840 shows the special case  A. x  T.. Theorem spi 1738 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 1534 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
 |-  ph   =>    |- 
 A. x A. y ph
 
Theoremmpg 1535 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
 |-  ( A. x ph  ->  ps )   &    |-  ph   =>    |- 
 ps
 
Theoremmpgbi 1536 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 1537 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 1538 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 1539 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 1540 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- 
 F/ x ph
 
Theoremnftru 1541 The true constant has no free variables. (This can also be proven in one step with nfv 1605, but this proof does not use ax-17 1603.) (Contributed by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x  T.
 
Theoremnex 1542 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
 |- 
 -.  ph   =>    |- 
 -.  E. x ph
 
Theoremnfnth 1543 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 1544 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 1545 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 1546 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theorem2alimi 1547 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 1548 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 1549 Variant of al2imi 1548 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x ph 
 /\  A. x ps )  ->  A. x ch )
 
Theoremalimdh 1550 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 1551 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( A. x ph  <->  A. x ps )
 )
 
Theoremalrimih 1552 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 1553 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 1554 Inference adding 2 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 1555 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2386 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfbii 1556 Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  <->  ps )   =>    |-  ( F/ x ph  <->  F/ x ps )
 
Theoremnfxfr 1557 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 1558 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 1559 Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  <->  -.  E. x  -.  ph )
 
Theorem2nalexn 1560 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 1561 Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x  -.  ph  <->  -. 
 A. x ph )
 
Theoremexim 1562 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 1563 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  E. x ps )
 
Theorem2eximi 1564 Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  E. x E. y ps )
 
Theoremalinexa 1565 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
 |-  ( A. x (
 ph  ->  -.  ps )  <->  -. 
 E. x ( ph  /\ 
 ps ) )
 
Theoremalexn 1566 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
 |-  ( A. x E. y  -.  ph  <->  -.  E. x A. y ph )
 
Theorem2exnexn 1567 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 1568 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremexbii 1569 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 1570 Inference adding 2 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 1571 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 1572 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 1573 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  E. x ( ps  /\  ph ) )
 
Theoremalrimdh 1574 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 1575 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 1576 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremalbidh 1577 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 1578 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 1579 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 1580 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 ) )
 
Theorem19.26-2 1581 Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
 |-  ( A. x A. y ( ph  /\  ps ) 
 <->  ( A. x A. y ph  /\  A. x A. y ps ) )
 
Theorem19.26-3an 1582 Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
 |-  ( A. x (
 ph  /\  ps  /\  ch ) 
 <->  ( A. x ph  /\ 
 A. x ps  /\  A. x ch ) )
 
Theorem19.29 1583 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( A. x ph 
 /\  E. x ps )  ->  E. x ( ph  /\ 
 ps ) )
 
Theorem19.29r 1584 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
 |-  ( ( E. x ph 
 /\  A. x ps )  ->  E. x ( ph  /\ 
 ps ) )
 
Theorem19.29r2 1585 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
 |-  ( ( E. x E. y ph  /\  A. x A. y ps )  ->  E. x E. y
 ( ph  /\  ps )
 )
 
Theorem19.29x 1586 Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
 |-  ( ( E. x A. y ph  /\  A. x E. y ps )  ->  E. x E. y
 ( ph  /\  ps )
 )
 
Theorem19.35 1587 Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( A. x ph 
 ->  E. x ps )
 )
 
Theorem19.35i 1588 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  E. x ps )
 
Theorem19.35ri 1589 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  E. x ps )   =>    |-  E. x ( ph  ->  ps )
 
Theorem19.25 1590 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. y E. x ( ph  ->  ps )  ->  ( E. y A. x ph  ->  E. y E. x ps ) )
 
Theorem19.30 1591 Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x (
 ph  \/  ps )  ->  ( A. x ph  \/  E. x ps )
 )
 
Theorem19.43 1592 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.43OLD 1593 Obsolete proof of 19.43 1592 as of 3-May-2016. Leave this in for the example on the mmrecent.html page. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.33 1594 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( A. x ph 
 \/  A. x ps )  ->  A. x ( ph  \/  ps ) )
 
Theorem19.33b 1595 The antecedent provides a condition implying the converse of 19.33 1594. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
 |-  ( -.  ( E. x ph  /\  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph 
 \/  A. x ps )
 ) )
 
Theorem19.40 1596 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  ->  ( E. x ph  /\ 
 E. x ps )
 )
 
Theorem19.40-2 1597 Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ph  /\  ps )  ->  ( E. x E. y ph  /\  E. x E. y ps )
 )
 
Theoremalbiim 1598 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps 
 ->  ph ) ) )
 
Theorem2albiim 1599 Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)
 |-  ( A. x A. y ( ph  <->  ps )  <->  ( A. x A. y ( ph  ->  ps )  /\  A. x A. y ( ps  ->  ph ) ) )
 
Theoremexintrbi 1600 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  <->  E. x ( ph  /\ 
 ps ) ) )
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