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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"), ("is a member of"), and ("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

The inference rules in this section will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic - please start with the statement wn 3 instead.

Theoremidi 1 (Note: This inference rule and the next one, a1ii 2, will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic; please start with the statement wn 3 instead.)

This inference says "if 𝜑 is true then 𝜑 is true". This inference requires no axioms for its proof, and 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. You can remove this using the metamath-exe (Metamath program) Proof Assistant using the "MM-PA> MINIMIZE_WITH *" command. This is the inference associated with id 22, hence its name. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑       𝜑

Theorema1ii 2 (Note: This inference rule and the previous one, idi 1, will normally never appear in a completed proof.)

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 (Metamath-exe) program Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user 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 "MM-PA> ASSIGN LAST" can be used. This step will be replicated in hypothesis a1ii.1, from where the development of the main proof can continue. (2) Develop the independent subproof backwards from hypothesis a1ii.2. If desired, use a "MM-PA> LET STEP" command to pre-assign the conclusion of the independent subproof to a1ii.2. (3) After the independent subproof is complete, use "MM-PA> 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 "MM-PA> MINIMIZE_WITH *" to clean up (discard) all a1ii 2 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. It is the inference associated with a1i 11. (Contributed by NM, 7-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑    &   𝜓       𝜑

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 (𝜑𝜑), 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 22).

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 6, ax-2 7, ax-3 8, and (for modus ponens) ax-mp 5. 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 890, 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 mmbiblio.html 890 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 𝜑 is a wff, so is ¬ 𝜑 or "not 𝜑". 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 𝜑 is true, then ¬ 𝜑 is false; if 𝜑 is false, then ¬ 𝜑 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 1957 and wel 2108).
wff ¬ 𝜑

Syntaxwi 4 If 𝜑 and 𝜓 are wff's, so is (𝜑𝜓) or "𝜑 implies 𝜓". Part of the recursive definition of a wff. The resulting wff is (interpreted as) false when 𝜑 is true and 𝜓 is false; it is true otherwise. Think of the truth table for an OR gate with input 𝜑 connected through an inverter. After we state the axioms of propositional calculus (ax-1 6, ax-2 7, ax-3 8, and ax-mp 5) and define the biconditional (df-bi 209), the constant true (df-tru 1533), and the constant false (df-fal 1543), we will be able to prove these truth table values: ((⊤ → ⊤) ↔ ⊤) (truimtru 1553), ((⊤ → ⊥) ↔ ⊥) (truimfal 1554), ((⊥ → ⊤) ↔ ⊤) (falimtru 1555), and ((⊥ → ⊥) ↔ ⊤) (falimfal 1556). These have straightforward meanings, for example, ((⊤ → ⊤) ↔ ⊤) just means "the value of (⊤ → ⊤) is ".

The left-hand wff is called the antecedent, and the right-hand wff is called the consequent. In the case of (𝜑 → (𝜓𝜒)), the middle 𝜓 may be informally called either an antecedent or part of the consequent depending on context. Contrast with (df-bi 209), (df-an 399), and (df-or 844).

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 is 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 119). In short, material implication has a very specific technical definition, and misunderstandings of it are sometimes called "paradoxes of logical implication".

wff (𝜑𝜓)

1.2.2  The axioms of propositional calculus

Propositional calculus (axioms ax-1 6 through ax-3 8 and rule ax-mp 5) 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 5) the wffs ax-1 6, ax-2 7, pm2.04 90, con3 156, notnot 144, and notnotr 132. 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 90) and replacing the last three with our ax-3 8. (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-mp 5 Rule of Modus Ponens. The postulated inference rule of propositional calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if 𝜑 is true, and 𝜑 implies 𝜓, then 𝜓 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 mode that by affirming affirms" - remark in [Sanford] p. 39. This rule is similar to the rule of modus tollens mto 199.

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, 30-Sep-1992.)

𝜑    &   (𝜑𝜓)       𝜓

Axiomax-1 6 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 𝜑 and 𝜓 to the assertion of 𝜑 simply". It is Proposition 1 of [Frege1879] p. 26, its first axiom. (Contributed by NM, 30-Sep-1992.)
(𝜑 → (𝜓𝜑))

Axiomax-2 7 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; see Proposition 2 of [Frege1879] p. 26. 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 pm5.41 394. (Contributed by NM, 30-Sep-1992.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Axiomax-3 8 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, 30-Sep-1992.) Use its alias con4 113 instead. (New usage is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

1.2.3  Logical implication

The results in this section are based on implication only, and avoid ax-3 8, so are intuitionistic. The system { ax-mp 5, ax-1 6, ax-2 7 } axiomatizes what is sometimes called "intuitionistic implicational calculus" or "minimal implicational calculus".

In an implication, the wff before the arrow is called the "antecedent" and the wff after the arrow is called the "consequent".

Theoremmp2 9 A double modus ponens inference. (Contributed by NM, 5-Apr-1994.)
𝜑    &   𝜓    &   (𝜑 → (𝜓𝜒))       𝜒

Theoremmp2b 10 A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.)
𝜑    &   (𝜑𝜓)    &   (𝜓𝜒)       𝜒

Theorema1i 11 Inference introducing an antecedent. Inference associated with ax-1 6. Its associated inference is a1ii 2. See conventions 28171 for a definition of "associated inference". (Contributed by NM, 29-Dec-1992.)
𝜑       (𝜓𝜑)

Theorem2a1i 12 Inference introducing two antecedents. Two applications of a1i 11. Inference associated with 2a1 28. (Contributed by Jeff Hankins, 4-Aug-2009.)
𝜑       (𝜓 → (𝜒𝜑))

Theoremmp1i 13 Inference detaching an antecedent and introducing a new one. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝜑    &   (𝜑𝜓)       (𝜒𝜓)

Theorema2i 14 Inference distributing an antecedent. Inference associated with ax-2 7. Its associated inference is mpd 15. (Contributed by NM, 29-Dec-1992.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜑𝜒))

Theoremmpd 15 A modus ponens deduction. A translation of natural deduction rule E ( elimination), see natded 28174. Deduction form of ax-mp 5. Inference associated with a2i 14. Commuted form of mpcom 38. (Contributed by NM, 29-Dec-1992.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremimim2i 16 Inference adding common antecedents in an implication. Inference associated with imim2 58. Its associated inference is syl 17. (Contributed by NM, 28-Dec-1992.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremsyl 17 An inference version of the transitive laws for implication imim2 58 and imim1 83 (and imim1i 63 and imim2i 16), which Russell and Whitehead call "the principle of the syllogism ... because ... the syllogism in Barbara is derived from [syl 17]" (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Some authors call this law a "hypothetical syllogism". Its associated inference is mp2b 10.

(A bit of trivia: this is the most commonly referenced assertion in our database (13449 times as of 22-Jul-2021). In second place is eqid 2819 (9597 times), followed by adantr 483 (8861 times), syl2anc 586 (7421 times), adantl 484 (6403 times), and simpr 487 (5829 times). The Metamath program command 'show usage' shows the number of references.)

(Contributed by NM, 30-Sep-1992.) (Proof shortened by Mel L. O'Cat, 20-Oct-2011.) (Proof shortened by Wolf Lammen, 26-Jul-2012.)

(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theorem3syl 18 Inference chaining two syllogisms syl 17. Inference associated with imim12i 62. (Contributed by NM, 28-Dec-1992.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜑𝜃)

Theorem4syl 19 Inference chaining three syllogisms syl 17. (Contributed by BJ, 14-Jul-2018.) The use of this theorem is marked "discouraged" because it can cause the Metamath program "MM-PA> MINIMIZE_WITH *" command to have very long run times. However, feel free to use "MM-PA> MINIMIZE_WITH 4syl / OVERRIDE" if you wish. Remember to update the "discouraged" file if it gets used. (New usage is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)    &   (𝜃𝜏)       (𝜑𝜏)

Theoremmpi 20 A nested modus ponens inference. Inference associated with com12 32. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremmpisyl 21 A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
(𝜑𝜓)    &   𝜒    &   (𝜓 → (𝜒𝜃))       (𝜑𝜃)

Theoremid 22 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For another version of the proof directly from axioms, see idALT 23. Its associated inference, idi 1, requires no axioms for its proof, contrary to id 22. Note that the second occurrences of 𝜑 in Steps 1 and 2 may be simultaneously replaced by any wff 𝜓, which may ease the understanding of the proof. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
(𝜑𝜑)

TheoremidALT 23 Alternate proof of id 22. 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.appstate.edu/~hirstjl/primer/hirst.pdf 22. Note that the second occurrences of 𝜑 in Steps 1 to 4 and the sixth in Step 3 may be simultaneously replaced by any wff 𝜓, which may ease the understanding of the proof. For a shorter version of the proof that takes advantage of previously proved theorems, see id 22. (Contributed by NM, 30-Sep-1992.) (Proof modification is discouraged.) Use id 22 instead. (New usage is discouraged.)
(𝜑𝜑)

Theoremidd 24 Principle of identity id 22 with antecedent. (Contributed by NM, 26-Nov-1995.)
(𝜑 → (𝜓𝜓))

Theorema1d 25 Deduction introducing an embedded antecedent. Deduction form of ax-1 6 and a1i 11. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))

Theorem2a1d 26 Deduction introducing two antecedents. Two applications of a1d 25. Deduction associated with 2a1 28 and 2a1i 12. (Contributed by BJ, 10-Aug-2020.)
(𝜑𝜓)       (𝜑 → (𝜒 → (𝜃𝜓)))

Theorema1i13 27 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜓𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theorem2a1 28 A double form of ax-1 6. Its associated inference is 2a1i 12. Its associated deduction is 2a1d 26. (Contributed by BJ, 10-Aug-2020.) (Proof shortened by Wolf Lammen, 1-Sep-2020.)
(𝜑 → (𝜓 → (𝜒𝜑)))

Theorema2d 29 Deduction distributing an embedded antecedent. Deduction form of ax-2 7. (Contributed by NM, 23-Jun-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))

Theoremsylcom 30 Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜓 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theoremsyl5com 31 Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜑 → (𝜒𝜃))

Theoremcom12 32 Inference that swaps (commutes) antecedents in an implication. Inference associated with pm2.04 90. Its associated inference is mpi 20. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
(𝜑 → (𝜓𝜒))       (𝜓 → (𝜑𝜒))

Theoremsyl11 33 A syllogism inference. Commuted form of an instance of syl 17. (Contributed by BJ, 25-Oct-2021.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜑)       (𝜓 → (𝜃𝜒))

Theoremsyl5 34 A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl6 35 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl56 36 Combine syl5 34 and syl6 35. (Contributed by NM, 14-Nov-2013.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))    &   (𝜃𝜏)       (𝜒 → (𝜑𝜏))

Theoremsyl6com 37 Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜓 → (𝜑𝜃))

Theoremmpcom 38 Modus ponens inference with commutation of antecedents. Commuted form of mpd 15. (Contributed by NM, 17-Mar-1996.)
(𝜓𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜓𝜒)

Theoremsyli 39 Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)
(𝜓 → (𝜑𝜒))    &   (𝜒 → (𝜑𝜃))       (𝜓 → (𝜑𝜃))

Theoremsyl2im 40 Replace two antecedents. Implication-only version of syl2an 597. (Contributed by Wolf Lammen, 14-May-2013.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜒𝜏))

Theoremsyl2imc 41 A commuted version of syl2im 40. Implication-only version of syl2anr 598. (Contributed by BJ, 20-Oct-2021.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓 → (𝜃𝜏))       (𝜒 → (𝜑𝜏))

Theorempm2.27 42 This theorem, sometimes called "Assertion" or "Pon" (for "ponens"), can be thought of as a closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. (Contributed by NM, 15-Jul-1993.)
(𝜑 → ((𝜑𝜓) → 𝜓))

Theoremmpdd 43 A nested modus ponens deduction. Double deduction associated with ax-mp 5. Deduction associated with mpd 15. (Contributed by NM, 12-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremmpid 44 A nested modus ponens deduction. Deduction associated with mpi 20. (Contributed by NM, 14-Dec-2004.)
(𝜑𝜒)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremmpdi 45 A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.)
(𝜓𝜒)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremmpii 46 A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)
𝜒    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremsyld 47 Syllogism deduction. Deduction associated with syl 17. See conventions 28171 for the meaning of "associated deduction" or "deduction form". (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theoremsyldc 48 Syllogism deduction. Commuted form of syld 47. (Contributed by BJ, 25-Oct-2021.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜓 → (𝜑𝜃))

Theoremmp2d 49 A double modus ponens deduction. Deduction associated with mp2 9. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑𝜃)

Theorema1dd 50 Double deduction introducing an antecedent. Deduction associated with a1d 25. Double deduction associated with ax-1 6 and a1i 11. (Contributed by NM, 17-Dec-2004.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theorem2a1dd 51 Double deduction introducing two antecedents. Two applications of 2a1dd 51. Deduction associated with 2a1d 26. Double deduction associated with 2a1 28 and 2a1i 12. (Contributed by Jeff Hankins, 5-Aug-2009.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜃 → (𝜏𝜒))))

Theorempm2.43i 52 Inference absorbing redundant antecedent. Inference associated with pm2.43 56. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
(𝜑 → (𝜑𝜓))       (𝜑𝜓)

Theorempm2.43d 53 Deduction absorbing redundant antecedent. Deduction associated with pm2.43 56 and pm2.43i 52. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
(𝜑 → (𝜓 → (𝜓𝜒)))       (𝜑 → (𝜓𝜒))

Theorempm2.43a 54 Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
(𝜓 → (𝜑 → (𝜓𝜒)))       (𝜓 → (𝜑𝜒))

Theorempm2.43b 55 Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.)
(𝜓 → (𝜑 → (𝜓𝜒)))       (𝜑 → (𝜓𝜒))

Theorempm2.43 56 Absorption of redundant antecedent. Also called the "Contraction" or "Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

Theoremimim2d 57 Deduction adding nested antecedents. Deduction associated with imim2 58 and imim2i 16. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))

Theoremimim2 58 A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Its associated inference is imim2i 16. Its associated deduction is imim2d 57. An alternate proof from more basic results is given by ax-1 6 followed by a2d 29. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 6-Sep-2012.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theoremembantd 59 Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theorem3syld 60 Triple syllogism deduction. Deduction associated with 3syld 60. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremsylsyld 61 A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜒𝜏))

Theoremimim12i 62 Inference joining two implications. Inference associated with imim12 105. Its associated inference is 3syl 18. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Mel L. O'Cat, 29-Oct-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜓𝜒) → (𝜑𝜃))

Theoremimim1i 63 Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Inference associated with imim1 83. Its associated inference is syl 17. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
(𝜑𝜓)       ((𝜓𝜒) → (𝜑𝜒))

Theoremimim3i 64 Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.)
(𝜑 → (𝜓𝜒))       ((𝜃𝜑) → ((𝜃𝜓) → (𝜃𝜒)))

Theoremsylc 65 A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓 → (𝜒𝜃))       (𝜑𝜃)

Theoremsyl3c 66 A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (𝜑𝜏)

Theoremsyl6mpi 67 A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   (𝜒 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremmpsyl 68 Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
𝜑    &   (𝜓𝜒)    &   (𝜑 → (𝜒𝜃))       (𝜓𝜃)

Theoremmpsylsyld 69 Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜓 → (𝜒𝜏))

Theoremsyl6c 70 Inference combining syl6 35 with contraction. (Contributed by Alan Sare, 2-May-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜒 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremsyl6ci 71 A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   (𝜒 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremsyldd 72 Nested syllogism deduction. Deduction associated with syld 47. Double deduction associated with syl 17. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜃𝜏)))       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremsyl5d 73 A nested syllogism deduction. Deduction associated with syl5 34. (Contributed by NM, 14-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃 → (𝜒𝜏)))       (𝜑 → (𝜃 → (𝜓𝜏)))

Theoremsyl7 74 A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 12-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
(𝜑𝜓)    &   (𝜒 → (𝜃 → (𝜓𝜏)))       (𝜒 → (𝜃 → (𝜑𝜏)))

Theoremsyl6d 75 A nested syllogism deduction. Deduction associated with syl6 35. (Contributed by NM, 11-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremsyl8 76 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.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜃𝜏)       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremsyl9 77 A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       (𝜑 → (𝜃 → (𝜓𝜏)))

Theoremsyl9r 78 A nested syllogism inference with different antecedents. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       (𝜃 → (𝜑 → (𝜓𝜏)))

Theoremsyl10 79 A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   (𝜒 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜃𝜂)))

Theorema1ddd 80 Triple deduction introducing an antecedent to a wff. Deduction associated with a1dd 50. Double deduction associated with a1d 25. Triple deduction associated with ax-1 6 and a1i 11. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜑 → (𝜓 → (𝜒𝜏)))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremimim12d 81 Deduction combining antecedents and consequents. Deduction associated with imim12 105 and imim12i 62. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mel L. O'Cat, 30-Oct-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜒𝜃) → (𝜓𝜏)))

Theoremimim1d 82 Deduction adding nested consequents. Deduction associated with imim1 83 and imim1i 63. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜒𝜃) → (𝜓𝜃)))

Theoremimim1 83 A closed form of syllogism (see syl 17). Theorem *2.06 of [WhiteheadRussell] p. 100. Its associated inference is imim1i 63. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theorempm2.83 84 Theorem *2.83 of [WhiteheadRussell] p. 108. Closed form of syld 47. (Contributed by NM, 3-Jan-2005.)
((𝜑 → (𝜓𝜒)) → ((𝜑 → (𝜒𝜃)) → (𝜑 → (𝜓𝜃))))

Theorempeirceroll 85 Over minimal implicational calculus, Peirce's axiom peirce 204 implies an axiom sometimes called "Roll", (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → 𝜑)), of which looinv 205 is a special instance. The converse also holds: substitute (𝜑𝜓) for 𝜒 in Roll and use id 22 and ax-mp 5. (Contributed by BJ, 15-Jun-2021.)
((((𝜑𝜓) → 𝜑) → 𝜑) → (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → 𝜑)))

Theoremcom23 86 Commutation of antecedents. Swap 2nd and 3rd. Deduction associated with com12 32. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜒 → (𝜓𝜃)))

Theoremcom3r 87 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜒 → (𝜑 → (𝜓𝜃)))

Theoremcom13 88 Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜒 → (𝜓 → (𝜑𝜃)))

Theoremcom3l 89 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜓 → (𝜒 → (𝜑𝜃)))

Theorempm2.04 90 Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. This was the third axiom in Frege's logic system, specifically Proposition 8 of [Frege1879] p. 35. Its associated inference is com12 32. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremcom34 91 Commutation of antecedents. Swap 3rd and 4th. Deduction associated with com23 86. Double deduction associated with com12 32. (Contributed by NM, 25-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏))))

Theoremcom4l 92 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜓 → (𝜒 → (𝜃 → (𝜑𝜏))))

Theoremcom4t 93 Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜒 → (𝜃 → (𝜑 → (𝜓𝜏))))

Theoremcom4r 94 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜃 → (𝜑 → (𝜓 → (𝜒𝜏))))

Theoremcom24 95 Commutation of antecedents. Swap 2nd and 4th. Deduction associated with com13 88. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜃 → (𝜒 → (𝜓𝜏))))

Theoremcom14 96 Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜃 → (𝜓 → (𝜒 → (𝜑𝜏))))

Theoremcom45 97 Commutation of antecedents. Swap 4th and 5th. Deduction associated with com34 91. Double deduction associated with com23 86. Triple deduction associated with com12 32. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃𝜂)))))

Theoremcom35 98 Commutation of antecedents. Swap 3rd and 5th. Deduction associated with com24 95. Double deduction associated with com13 88. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜏 → (𝜃 → (𝜒𝜂)))))

Theoremcom25 99 Commutation of antecedents. Swap 2nd and 5th. Deduction associated with com14 96. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜏 → (𝜒 → (𝜃 → (𝜓𝜂)))))

Theoremcom5l 100 Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜓 → (𝜒 → (𝜃 → (𝜏 → (𝜑𝜂)))))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44891
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