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Theorem List for Metamath Proof Explorer - 1-100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY

Logic can be defined as the "study of the principles of correct reasoning" (Merrilee H. Salmon's 1991 "Informal Reasoning and Informal Logic" in Informal Reasoning and Education ) or as "a formal system using symbolic techniques and mathematical methods to establish truth-values" (the Oxford English Dictionary).

This section formally defines the logic system we will use. In particular, it defines symbols for declaring truthful statements, along with rules for deriving truthful statements from other truthful statements. The system defined here is classical first order logic with equality (the most common logic system used by mathematicians).

We begin with a few housekeeping items in pre-logic, and then introduce propositional calculus (both its axioms and important theorems that can be derived from them). Propositional calculus deals with general truths about well-formed formulas (wffs) regardless of how they are constructed. This is followed by proofs that other axiomatizations of classical propositional calculus can be derived from the axioms we have chosen to use.

We then define predicate calculus, which adds additional symbols and rules useful for discussing objects (beyond simply true or false). In particular, it introduces the symbols  = ("equals"),  e. ("is a member of"), and  A. ("for all"). The first two are called "predicates." A predicate specifies a true or false relationship between its two arguments.

 
1.1  Pre-logic

This section includes a few "housekeeping" mechanisms before we begin defining the basics of logic.

 
1.1.1  Inferences for assisting proof development
 
Theoremdummylink 1 (Note: This inference rule and the next one, idi 2, will normally never appear in a completed proof. It can be ignored if you are using this database to assist learning logic - please start with the statement wn 3 instead.)

This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step.

The metamath program's Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user to work on isolated subproofs. This inference provides a workaround for this limitation. It can be inserted at any point in a proof to allow an independent subproof to be developed on the side, for later use as part of the final proof.

Instructions: (1) Assign this inference to any unknown step in the proof. Typically, the last unknown step is the most convenient, since 'assign last' can be used. This step will be replicated in hypothesis dummylink.1, from where the development of the main proof can continue. (2) Develop the independent subproof backwards from hypothesis dummylink.2. If desired, use a 'let' command to pre-assign the conclusion of the independent subproof to dummylink.2. (3) After the independent subproof is complete, use 'improve all' to assign it automatically to an unknown step in the main proof that matches it. (4) After the entire proof is complete, use 'minimize *' to clean up (discard) all dummylink references automatically.

This inference was originally designed to assist importing partially completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, no axioms are required for its proof. (Contributed by NM, 7-Feb-2006.)

 |-  ph   &    |- 
 ps   =>    |-  ph
 
Theoremidi 2 Inference form of id 20. This inference rule, which requires no axioms for its proof, is useful as a copy-paste mechanism during proof development in mmj2. It is normally not referenced in the final version of a proof, since it is always redundant and can be removed using the 'minimize *' command in the metamath program's Proof Assistant. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ph   =>    |-  ph
 
1.2  Propositional calculus

Propositional calculus deals with general truths about well-formed formulas (wffs) regardless of how they are constructed. The simplest propositional truth is  ( ph  ->  ph ), which can be read "if something is true, then it is true" - rather trivial and obvious, but nonetheless it must be proved from the axioms (see theorem id 20).

Our system of propositional calculus consists of three basic axioms and another axiom that defines the modus-ponens inference rule. It is attributed to Jan Lukasiewicz (pronounced woo-kah-SHAY-vitch) and was popularized by Alonzo Church, who called it system P2. (Thanks to Ted Ulrich for this information.) These axioms are ax-1 5, ax-2 6, ax-3 7, and (for modus ponens) ax-mp 8. Some closely followed texts include [Margaris] for the axioms and [WhiteheadRussell] for the theorems.

The propositional calculus used here is the classical system widely used by mathematicians. In particular, this logic system accepts the "law of the excluded middle" as proven in exmid 405, which says that a logical statement is either true or not true. This is an essential distinction of classical logic and is not a theorem of intuitionistic logic.

All 194 axioms, definitions, and theorems for propositional calculus in Principia Mathematica (specifically *1.2 through *5.75) are axioms or formally proven. See the Bibliographic Cross-References at http://us.metamath.org/mpeuni/mmbiblio.html for a complete cross-reference from sources used to its formalization in the Metamath Proof Explorer.

 
1.2.1  Recursively define primitive wffs for propositional calculus
 
Syntaxwn 3 If  ph is a wff, so is  -.  ph or "not  ph." Part of the recursive definition of a wff (well-formed formula). In classical logic (which is our logic), a wff is interpreted as either true or false. So if  ph is true, then  -.  ph is false; if  ph is false, then  -.  ph is true. Traditionally, Greek letters are used to represent wffs, and we follow this convention. In propositional calculus, we define only wffs built up from other wffs, i.e. there is no starting or "atomic" wff. Later, in predicate calculus, we will extend the basic wff definition by including atomic wffs (weq 1650 and wel 1722).
 wff  -.  ph
 
Syntaxwi 4 If  ph and  ps are wff's, so is  ( ph  ->  ps ) or " ph implies  ps." Part of the recursive definition of a wff. The resulting wff is (interpreted as) false when  ph is true and 
ps is false; it is true otherwise. Think of the truth table for an OR gate with input  ph connected through an inverter. After we define the axioms of propositional calculus (ax-1 5, ax-2 6, ax-3 7, and ax-mp 8), the biconditional (df-bi 178), the constant true  T. (df-tru 1325), and the constant false  F. (df-fal 1326), we will be able to prove these truth table values:  ( (  T. 
->  T.  )  <->  T.  ) (truimtru 1350),  ( (  T.  ->  F.  )  <->  F.  ) (truimfal 1351), 
( (  F.  ->  T.  )  <->  T.  ) (falimtru 1352), and  ( (  F.  ->  F.  )  <->  T.  ) (falimfal 1353). These have straightforward meanings, for example,  ( (  T.  ->  T.  ) 
<->  T.  ) just means "the value of 
T.  ->  T. is  T.".

The left-hand wff is called the antecedent, and the right-hand wff is called the consequent. In the case of  ( ph  ->  ( ps  ->  ch ) ), the middle  ps may be informally called either an antecedent or part of the consequent depending on context. Contrast with  <-> (df-bi 178),  /\ (df-an 361), and  \/ (df-or 360).

This is called "material implication" and the arrow is usually read as "implies." However, material implication is not identical to the meaning of "implies" in natural language. For example, the word "implies" may suggest a causal relationship in natural language. Material implication does not require any causal relationship. Also, note that in material implication, if the consequent is true then the wff is always true (even if the antecedent is false). Thus, if "implies" means material implication, it is true that "if the moon is made of green cheese that implies that 5=5" (because 5=5). Similarly, if the antecedent is false, the wff is always true. Thus, it is true that, "if the moon made of green cheese that implies that 5=7" (because the moon is not actually made of green cheese). A contradiction implies anything (pm2.21i 125). In short, material implication has a very specific technical definition, and misunderstandings of it are sometimes called "paradoxes of logical implication."

 wff  ( ph  ->  ps )
 
1.2.2  The axioms of propositional calculus

Postulate the three axioms of classical propositional calculus.

Propositional calculus (axioms ax-1 5 through ax-3 7 and rule ax-mp 8) can be thought of as asserting formulas that are universally "true" when their variables are replaced by any combination of "true" and "false." Propositional calculus was first formalized by Frege in 1879, using as his axioms (in addition to rule ax-mp 8) the wffs ax-1 5, ax-2 6, pm2.04 78, con3 128, notnot2 106, and notnot1 116. Around 1930, Lukasiewicz simplified the system by eliminating the third (which follows from the first two, as you can see by looking at the proof of pm2.04 78) and replacing the last three with our ax-3 7. (Thanks to Ted Ulrich for this information.)

The theorems of propositional calculus are also called tautologies. Tautologies can be proved very simply using truth tables, based on the true/false interpretation of propositional calculus. To do this, we assign all possible combinations of true and false to the wff variables and verify that the result (using the rules described in wi 4 and wn 3) always evaluates to true. This is called the semantic approach. Our approach is called the syntactic approach, in which everything is derived from axioms. A metatheorem called the Completeness Theorem for Propositional Calculus shows that the two approaches are equivalent and even provides an algorithm for automatically generating syntactic proofs from a truth table. Those proofs, however, tend to be long, since truth tables grow exponentially with the number of variables, and the much shorter proofs that we show here were found manually.

 
Axiomax-1 5 Axiom Simp. Axiom A1 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. The 3 axioms are also given as Definition 2.1 of [Hamilton] p. 28. This axiom is called Simp or "the principle of simplification" in Principia Mathematica (Theorem *2.02 of [WhiteheadRussell] p. 100) because "it enables us to pass from the joint assertion of  ph and  ps to the assertion of  ph simply." (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Axiomax-2 6 Axiom Frege. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as Frege in the literature. It is also proved as Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by pm5.41 354. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Axiomax-3 7 Axiom Transp. Axiom A3 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This axiom is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103). We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( -.  ph  ->  -.  ps )  ->  ( ps  ->  ph ) )
 
Axiomax-mp 8 Rule of Modus Ponens. The postulated inference rule of propositional calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if  ph is true, and  ph implies  ps, then  ps must also be true." This rule is sometimes called "detachment," since it detaches the minor premise from the major premise. "Modus ponens" is short for "modus ponendo ponens," a Latin phrase that means "the mood that by affirming affirms" [Sanford] p. 39. This rule is similar to the rule of modus tollens mto 169.

Note: In some web page displays such as the Statement List, the symbols "&" and "=>" informally indicate the relationship between the hypotheses and the assertion (conclusion), abbreviating the English words "and" and "implies." They are not part of the formal language. (Contributed by NM, 5-Aug-1993.)

 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  ps
 
1.2.3  Logical implication

The results in this section are based on implication only, and avoid ax-3, so are intuitionistic. In an implication, the wff before the arrow is called the "antecedent" and the wff after the arrow is called the "consequent."

We will use the following descriptive terms very loosely: A "closed form" or "tautology" has no $e hypotheses. An "inference" has one or more $e hypotheses. A "deduction" is an inference in which the hypotheses and the conclusion share the same antecedent.

 
Theoremmp2 9 A double modus ponens inference. See mp2ALT 18 for a shorter proof using two more axioms. (Contributed by NM, 5-Apr-1994.) (Proof modification is discouraged.)
 |-  ph   &    |- 
 ps   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ch
 
Theoremmp2b 10 A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.)
 |-  ph   &    |-  ( ph  ->  ps )   &    |-  ( ps  ->  ch )   =>    |- 
 ch
 
Theorema1i 11 Inference derived from axiom ax-1 5. See a1d 23 for an explanation of our informal use of the terms "inference" and "deduction." See also the comment in syld 42. (Contributed by NM, 5-Aug-1993.)
 |-  ph   =>    |-  ( ps  ->  ph )
 
Theoremmp1i 12 Drop and replace an antecedent. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  ( ch  ->  ps )
 
Theorema2i 13 Inference derived from axiom ax-2 6. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )
 
Theoremimim2i 14 Inference adding common antecedents in an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  ->  ph )  ->  ( ch  ->  ps )
 )
 
Theoremmpd 15 A modus ponens deduction. A translation of natural deduction rule  -> E ( -> elimination), see natded 21664. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ch )
 
Theoremsyl 16 An inference version of the transitive laws for implication imim2 51 and imim1 72, which Russell and Whitehead call "the principle of the syllogism...because...the syllogism in Barbara is derived from them" (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Some authors call this law a "hypothetical syllogism."

(A bit of trivia: this is the most commonly referenced assertion in our database. In second place is eqid 2404, followed by syl2anc 643, adantr 452, syl3anc 1184, and ax-mp 8. The Metamath program command 'show usage' shows the number of references.) (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 20-Oct-2011.) (Proof shortened by Wolf Lammen, 26-Jul-2012.)

 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremmpi 17 A nested modus ponens inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
 |- 
 ps   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ch )
 
Theoremmp2ALT 18 Alternate proof of mp2 9 (shorter but uses two more axioms). (Contributed by Wolf Lammen, 23-Jul-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ph   &    |- 
 ps   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ch
 
Theorem3syl 19 Inference chaining two syllogisms. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ch )   &    |-  ( ch  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremid 20 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For another version of the proof directly from axioms, see id1 21. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
 |-  ( ph  ->  ph )
 
Theoremid1 21 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. This version is proved directly from the axioms for demonstration purposes. This proof is a popular example in the literature and is identical, step for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a) of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of [Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's A Primer for Logic and Proof p. 17 (PDF p. 23) at http://www.mathsci.appstate.edu/~hirstjl/primer/hirst.pdf. For a shorter version of the proof that takes advantage of previously proved theorems, see id 20. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  ph )
 
Theoremidd 22 Principle of identity with antecedent. (Contributed by NM, 26-Nov-1995.)
 |-  ( ph  ->  ( ps  ->  ps ) )
 
Theorema1d 23 Deduction introducing an embedded antecedent.

Naming convention: We often call a theorem a "deduction" and suffix its label with "d" whenever the hypotheses and conclusion are each prefixed with the same antecedent. This allows us to use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem would be used; here  ph would be replaced with a conjunction (df-an 361) of the hypotheses of the would-be deduction. By contrast, we tend to call the simpler version with no common antecedent an "inference" and suffix its label with "i"; compare theorem a1i 11. Finally, a "theorem" would be the form with no hypotheses; in this case the "theorem" form would be the original axiom ax-1 5. We usually show the theorem form without a suffix on its label (e.g. pm2.43 49 vs. pm2.43i 45 vs. pm2.43d 46). When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the theorem form with "g" (for "more general") as in uniex 4664 vs. uniexg 4665. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)

 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  ->  ps ) )
 
Theorema2d 24 Deduction distributing an embedded antecedent. (Contributed by NM, 23-Jun-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  ->  ch )  ->  ( ps  ->  th )
 ) )
 
Theorema1ii 25 Add two antecedents to a wff. See a1iiALT 26 for a shorter proof using one more axiom. (Contributed by Jeff Hankins, 4-Aug-2009.) (Proof modification is discouraged.)
 |- 
 ch   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theorema1iiALT 26 Alternate proof of a1ii 25 (shorter but uses one more axiom). (Contributed by Wolf Lammen, 23-Jul-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 ch   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremsylcom 27 Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ps  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremsyl5com 28 Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ( ps  ->  th ) )   =>    |-  ( ph  ->  ( ch  ->  th ) )
 
Theoremcom12 29 Inference that swaps (commutes) antecedents in an implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ps  ->  (
 ph  ->  ch ) )
 
Theoremsyl5 30 A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-May-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ( ps  ->  th ) )   =>    |-  ( ch  ->  ( ph  ->  th ) )
 
Theoremsyl6 31 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ch  ->  th )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremsyl56 32 Combine syl5 30 and syl6 31. (Contributed by NM, 14-Nov-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ( ps  ->  th ) )   &    |-  ( th  ->  ta )   =>    |-  ( ch  ->  ( ph  ->  ta ) )
 
Theoremsyl6com 33 Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ch  ->  th )   =>    |-  ( ps  ->  ( ph  ->  th ) )
 
Theoremmpcom 34 Modus ponens inference with commutation of antecedents. (Contributed by NM, 17-Mar-1996.)
 |-  ( ps  ->  ph )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ps  ->  ch )
 
Theoremsyli 35 Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)
 |-  ( ps  ->  ( ph  ->  ch ) )   &    |-  ( ch  ->  ( ph  ->  th ) )   =>    |-  ( ps  ->  ( ph  ->  th ) )
 
Theoremsyl2im 36 Replace two antecedents. Implication-only version of syl2an 464. (Contributed by Wolf Lammen, 14-May-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ps  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ch  ->  ta ) )
 
Theorempm2.27 37 This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 8. Theorem *2.27 of [WhiteheadRussell] p. 104. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  (
 ( ph  ->  ps )  ->  ps ) )
 
Theoremmpdd 38 A nested modus ponens deduction. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremmpid 39 A nested modus ponens deduction. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremmpdi 40 A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by O'Cat, 15-Jan-2008.)
 |-  ( ps  ->  ch )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremmpii 41 A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)
 |- 
 ch   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ps  ->  th )
 )
 
Theoremsyld 42 Syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

Notice that syld 42 has the same form as syl 16 with  ph added in front of each hypothesis and conclusion. When all theorems referenced in a proof are converted in this way, we can replace  ph with a hypothesis of the proof, allowing the hypothesis to be eliminated with id 20 and become an antecedent. The Deduction Theorem for propositional calculus, e.g. Theorem 3 in [Margaris] p. 56, tells us that this procedure is always possible.

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremmp2d 43 A double modus ponens deduction. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  th )
 
Theorema1dd 44 Deduction introducing a nested embedded antecedent. (Contributed by NM, 17-Dec-2004.) (Proof shortened by O'Cat, 15-Jan-2008.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
 
Theorempm2.43i 45 Inference absorbing redundant antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
 |-  ( ph  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  ps )
 
Theorempm2.43d 46 Deduction absorbing redundant antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
 |-  ( ph  ->  ( ps  ->  ( ps  ->  ch ) ) )   =>    |-  ( ph  ->  ( ps  ->  ch )
 )
 
Theorempm2.43a 47 Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by O'Cat, 28-Nov-2008.)
 |-  ( ps  ->  ( ph  ->  ( ps  ->  ch ) ) )   =>    |-  ( ps  ->  (
 ph  ->  ch ) )
 
Theorempm2.43b 48 Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.)
 |-  ( ps  ->  ( ph  ->  ( ps  ->  ch ) ) )   =>    |-  ( ph  ->  ( ps  ->  ch )
 )
 
Theorempm2.43 49 Absorption of redundant antecedent. Also called the "Contraction" or "Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 15-Aug-2004.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremimim2d 50 Deduction adding nested antecedents. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  ->  ps )  ->  ( th  ->  ch )
 ) )
 
Theoremimim2 51 A closed form of syllogism (see syl 16). Theorem *2.05 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Sep-2012.)
 |-  ( ( ph  ->  ps )  ->  ( ( ch  ->  ph )  ->  ( ch  ->  ps ) ) )
 
Theoremembantd 52 Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  th ) )
 
Theorem3syld 53 Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   &    |-  ( ph  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoremsylsyld 54 Virtual deduction rule e12 28545 without virtual deduction symbols. (Contributed by Alan Sare, 20-Apr-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th ) )   &    |-  ( ps  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ch  ->  ta ) )
 
Theoremimim12i 55 Inference joining two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 29-Oct-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   =>    |-  ( ( ps  ->  ch )  ->  ( ph  ->  th ) )
 
Theoremimim1i 56 Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ps  ->  ch )  ->  ( ph  ->  ch )
 )
 
Theoremimim3i 57 Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( th  -> 
 ph )  ->  (
 ( th  ->  ps )  ->  ( th  ->  ch )
 ) )
 
Theoremsylc 58 A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ps  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  th )
 
Theoremsyl3c 59 A syllogism inference combined with contraction. e111 28484 without virtual deductions. (Contributed by Alan Sare, 7-Jul-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ph  ->  ta )
 
Theoremsyl6mpi 60 e20 28548 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ( ch  ->  ( th  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoremmpsyl 61 Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ps  ->  th )
 
Theoremsyl6c 62 Inference combining syl6 31 with contraction. (Contributed by Alan Sare, 2-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ch  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoremsyldd 63 Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
Theoremsyl5d 64 A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat, 2-Feb-2006.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ( ch  ->  ta )
 ) )   =>    |-  ( ph  ->  ( th  ->  ( ps  ->  ta ) ) )
 
Theoremsyl7 65 A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ( th  ->  ( ps  ->  ta )
 ) )   =>    |-  ( ch  ->  ( th  ->  ( ph  ->  ta ) ) )
 
Theoremsyl6d 66 A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat, 2-Feb-2006.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
Theoremsyl8 67 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   &    |-  ( th  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
Theoremsyl9 68 A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( th  ->  ( ps  ->  ta ) ) )
 
Theoremsyl9r 69 A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ch  ->  ta ) )   =>    |-  ( th  ->  ( ph  ->  ( ps  ->  ta ) ) )
 
Theoremimim12d 70 Deduction combining antecedents and consequents. (Contributed by NM, 7-Aug-1994.) (Proof shortened by O'Cat, 30-Oct-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ph  ->  (
 ( ch  ->  th )  ->  ( ps  ->  ta )
 ) )
 
Theoremimim1d 71 Deduction adding nested consequents. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ch  ->  th )  ->  ( ps  ->  th )
 ) )
 
Theoremimim1 72 A closed form of syllogism (see syl 16). Theorem *2.06 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-May-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theorempm2.83 73 Theorem *2.83 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ( ph  ->  ( ch  ->  th ) )  ->  ( ph  ->  ( ps  ->  th ) ) ) )
 
Theoremcom23 74 Commutation of antecedents. Swap 2nd and 3rd. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )
 
Theoremcom3r 75 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ch  ->  (
 ph  ->  ( ps  ->  th ) ) )
 
Theoremcom13 76 Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ch  ->  ( ps  ->  ( ph  ->  th ) ) )
 
Theoremcom3l 77 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )
 
Theorempm2.04 78 Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ps  ->  ( ph  ->  ch )
 ) )
 
Theoremcom34 79 Commutation of antecedents. Swap 3rd and 4th. (Contributed by NM, 25-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ( ch  ->  ta )
 ) ) )
 
Theoremcom4l 80 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by O'Cat, 15-Aug-2004.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ps  ->  ( ch  ->  ( th  ->  ( ph  ->  ta )
 ) ) )
 
Theoremcom4t 81 Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ch  ->  ( th  ->  ( ph  ->  ( ps  ->  ta )
 ) ) )
 
Theoremcom4r 82 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( th  ->  (
 ph  ->  ( ps  ->  ( ch  ->  ta )
 ) ) )
 
Theoremcom24 83 Commutation of antecedents. Swap 2nd and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( th  ->  ( ch  ->  ( ps  ->  ta )
 ) ) )
 
Theoremcom14 84 Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( th  ->  ( ps  ->  ( ch  ->  ( ph  ->  ta )
 ) ) )
 
Theoremcom45 85 Commutation of antecedents. Swap 4th and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  ->  ( th  ->  et ) ) ) ) )
 
Theoremcom35 86 Commutation of antecedents. Swap 3rd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ta  ->  ( th  ->  ( ch  ->  et ) ) ) ) )
 
Theoremcom25 87 Commutation of antecedents. Swap 2nd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ph  ->  ( ta  ->  ( ch  ->  ( th  ->  ( ps  ->  et ) ) ) ) )
 
Theoremcom5l 88 Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  ( ph  ->  et ) ) ) ) )
 
Theoremcom15 89 Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ta  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ph  ->  et ) ) ) ) )
 
Theoremcom52l 90 Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ch  ->  ( th  ->  ( ta  ->  (
 ph  ->  ( ps  ->  et ) ) ) ) )
 
Theoremcom52r 91 Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( th  ->  ( ta  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )
 
Theoremcom5r 92 Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ta  ->  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  et ) ) ) ) )
 
Theoremjarr 93 Elimination of a nested antecedent as a kind of reversal of inference ja 155. (Contributed by Wolf Lammen, 9-May-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( ps  ->  ch )
 )
 
Theorempm2.86i 94 Inference based on pm2.86 96. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ch )
 )
 
Theorempm2.86d 95 Deduction based on pm2.86 96. (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
 |-  ( ph  ->  (
 ( ps  ->  ch )  ->  ( ps  ->  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theorempm2.86 96 Converse of axiom ax-2 6. Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  ->  ch ) ) )
 
Theoremloolin 97 The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. (Contributed by O'Cat, 12-Aug-2004.) (Proof modification is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ( ps  ->  ph ) )  ->  ( ps  ->  ph ) )
 
Theoremloowoz 98 An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, Reports on Mathematical Logic 10, 129-137 (1978). (Contributed by O'Cat, 8-Aug-2004.)
 |-  ( ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )  ->  ( ( ps  ->  ph )  ->  ( ps  ->  ch ) ) )
 
1.2.4  Logical negation

This section makes our first use of the third axiom of propositional calculus, ax-3 7.

 
Theoremcon4d 99 Deduction derived from axiom ax-3 7. (Contributed by NM, 26-Mar-1995.)
 |-  ( ph  ->  ( -.  ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  ( ch  ->  ps ) )
 
Theorempm2.21d 100 A contradiction implies anything. Deduction from pm2.21 102. (Contributed by NM, 10-Feb-1996.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
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