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**Statement List for Metamath Proof Explorer -** 101-200 - Page 2 of 175
Type	Label	Description
Statement

Theorem	notnotri 101	Inference from double negation.


Theorem	notnot1 102	Converse of double negation. Theorem *2.12 of [WhiteheadRussell] p. 101.


Theorem	notnoti 103	Infer double negation.


Theorem	pm2.01 104	Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. (The proof was shortened by O'Cat, 21-Nov-2008.


Theorem	pm2.01d 105	Deduction based on reductio ad absurdum.


Theorem	con2 106	Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100.


Theorem	con2d 107	A contraposition deduction.


Theorem	con1 108	Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102.


Theorem	con1d 109	A contraposition deduction.


Theorem	con3 110	Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103.


Theorem	con3d 111	A contraposition deduction.


Theorem	con1i 112	A contraposition inference. (The proof was shortened by O'Cat, 28-Nov-2008.)


Theorem	con2i 113	A contraposition inference. (The proof was shortened by O'Cat, 28-Nov-2008.)


Theorem	con3i 114	A contraposition inference.


Theorem	pm2.5 115	Theorem *2.5 of [WhiteheadRussell] p. 107.


Theorem	pm2.51 116	Theorem *2.51 of [WhiteheadRussell] p. 107.


Theorem	pm2.52 117	Theorem *2.52 of [WhiteheadRussell] p. 107.


Theorem	pm2.521 118	Theorem *2.521 of [WhiteheadRussell] p. 107.


Theorem	pm2.24i 119	Inference version of pm2.24 95.


Theorem	pm2.24d 120	Deduction version of pm2.21 92.


Theorem	mto 121	The rule of modus tollens.


Theorem	mtoi 122	Modus tollens inference.


Theorem	mtod 123	Modus tollens deduction.


Theorem	mt2 124	A rule similar to modus tollens.


Theorem	mt2i 125	Modus tollens inference.


Theorem	mt2d 126	Modus tollens deduction.


Theorem	mt3 127	A rule similar to modus tollens.


Theorem	mt3i 128	Modus tollens inference.


Theorem	mt3d 129	Modus tollens deduction.


Theorem	mt4d 130	Modus tollens deduction.


Theorem	nsyl 131	A negated syllogism inference.


Theorem	nsyld 132	A negated syllogism deduction.


Theorem	nsyl2 133	A negated syllogism inference.


Theorem	nsyl3 134	A negated syllogism inference.


Theorem	nsyl4 135	A negated syllogism inference.


Theorem	nsyli 136	A negated syllogism inference.


Theorem	pm3.2im 137	Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives. (The proof was shortened by Josh Purinton, 29-Dec-2000.)


Theorem	mth8 138	Theorem 8 of [Margaris] p. 60. (The proof was shortened by Josh Purinton, 29-Dec-2000.)


Theorem	pm2.61 139	Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an antecedent.


Theorem	pm2.61i 140	Inference eliminating an antecedent.


Theorem	pm2.61d 141	Deduction eliminating an antecedent.


Theorem	pm2.61d1 142	Inference eliminating an antecedent.


Theorem	pm2.61d2 143	Inference eliminating an antecedent.


Theorem	pm2.61ii 144	Inference eliminating two antecedents. (The proof was shortened by Josh Purinton, 29-Dec-2000.)


Theorem	pm2.61nii 145	Inference eliminating two antecedents. (The proof was shortened by Andrew Salmon, 25-May-2011.)


Theorem	pm2.61niiOLD 146	Inference eliminating two antecedents.


Theorem	pm2.61iii 147	Inference eliminating three antecedents.


Theorem	pm2.6 148	Theorem *2.6 of [WhiteheadRussell] p. 107.


Theorem	pm2.65 149	Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction.


Theorem	pm2.65i 150	Inference rule for proof by contradiction.


Theorem	pm2.65d 151	Deduction rule for proof by contradiction.


Theorem	ja 152	Inference joining the antecedents of two premises. (The proof was shortened by O'Cat, 19-Feb-2008.)


Theorem	jc 153	Inference joining the consequents of two premises.


Theorem	simplim 154	Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112.


Theorem	simprim 155	Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112.


Theorem	jad 156	Deduction form of ja 152. (Contributed by Scott Fenton, 13-Dec-2010.) (The proof was shortened by Andrew Salmon, 17-Sep-2011.)


Theorem	jadOLD2 157	Deduction form of ja 152. (Contributed by Scott Fenton, 13-Dec-2010.)


Theorem	impt 158	Importation theorem expressed with primitive connectives.


Theorem	expt 159	Exportation theorem expressed with primitive connectives.


Theorem	impi 160	An importation inference.


Theorem	expi 161	An exportation inference. (The proof was shortened by O'Cat, 28-Nov-2008.)


Theorem	bijust 162	Theorem used to justify definition of biconditional df-bi 164. (The proof was shortened by Josh Purinton, 29-Dec-2000.)


Logical equivalence

Syntax	wb 163	Extend our wff definition to include the biconditional connective.


Definition	df-bi 164	This is our first definition, which introduces and defines the biconditional connective . We define a wff of the form as an abbreviation for . Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as." Instead, we will later use the biconditional connective for this purpose (df-or 241 is its first use), as it allows us to use logic to manipulate definitions directly. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions. Of course, we cannot use this mechanism to define the biconditional itself, since it hasn't been introduced yet. Instead, we use a more general form of definition, described as follows. In its most general form, a definition is simply an assertion that introduces a new symbol (or a new combination of existing symbols, as in df-3an 860) that is eliminable and does not strengthen the existing language. The latter requirement means that the set of provable statements not containing the new symbol (or new combination) should remain exactly the same after the definition is introduced. Our definition of the biconditional may look unusual compared to most definitions, but it strictly satisfies these requirements. The justification for our definition is that if we mechanically replace the first wff above (the definiendum i.e. the thing being defined) with the second (the definiens i.e. the defining expression) in the definition, the definition becomes a substitution instance of previously proved theorem bijust 162. It is impossible to use df-bi 164 to prove any statement expressed in the original language that can't be proved from the original axioms. For if it were, we could replace it with instances of bijust 162 throughout the proof, thus obtaining a proof from the original axioms (contradiction). Note that from Metamath's point of view, a definition is just another axiom - i.e. an assertion we claim to be true - but from our high level point of view, we are are not strengthening the language. To indicate this fact, we prefix definition labels with "df-" instead of "ax-". (This prefixing is an informal convention that means nothing to the Metamath proof verifier; it is just for human readability.) See dfbi1 175, dfbi2 572, and dfbi3 733 for theorems suggesting typical textbook definitions of , showing that our definition has the properties we expect. Theorem dfbi 573 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.


Theorem	bi1 165	Property of the biconditional connective.


Theorem	bi2 166	Property of the biconditional connective.


Theorem	bi3 167	Property of the biconditional connective.


Theorem	biimpi 168	Infer an implication from a logical equivalence.


Theorem	biimpri 169	Infer a converse implication from a logical equivalence.


Theorem	biimpd 170	Deduce an implication from a logical equivalence.


Theorem	biimprd 171	Deduce a converse implication from a logical equivalence.


Theorem	biimpcd 172	Deduce a commuted implication from a logical equivalence.


Theorem	biimprcd 173	Deduce a converse commuted implication from a logical equivalence.


Theorem	impbii 174	Infer an equivalence from an implication and its converse.


Theorem	dfbi1 175	Relate the biconditional connective to primitive connectives. See dfbi1gb 176 for an unusual version proved directly from axioms.


Theorem	dfbi1gb 176	This proof of dfbi1 175, discovered by Gregory Bush on 8-Mar-2004, has several curious properties. First, it has only 17 steps directly from the axioms and df-bi 164, compared to over 800 steps were the proof of dfbi1 175 expanded into axioms. Second, step 2 demands only the property of "true"; any axiom (or theorem) could be used. It might be thought, therefore, that it is in some sense redundant, but in fact no proof is shorter than this (measured by number of steps). Third, it illustrates how intermediate steps can "blow up" in size even in short proofs. Fourth, the compressed proof is only 182 bytes (or 17 bytes in D-proof notation), but the generated web page is over 200kB with intermediate steps that are essentially incomprehensible to humans (other than Gregory Bush). If there were an obfuscated code contest for proofs, this would be a contender.


Theorem	bi2.04 177	Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122.


Theorem	notnot 178	Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117.


Theorem	pm4.8 179	Theorem *4.8 of [WhiteheadRussell] p. 122.


Theorem	pm4.81 180	Theorem *4.81 of [WhiteheadRussell] p. 122.


Theorem	con1b 181	Contraposition. Bidirectional version of con1 108.


Theorem	con2b 182	Contraposition. Bidirectional version of con2 106.


Theorem	con34b 183	Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116.


Theorem	pm5.4 184	Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125.


Theorem	imdi 185	Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125.


Theorem	pm5.41 186	Theorem *5.41 of [WhiteheadRussell] p. 125.


Theorem	biid 187	Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117.


Theorem	biidd 188	Principle of identity with antecedent.


Theorem	bicomi 189	Inference from commutative law for logical equivalence.


Theorem	bitri 190	An inference from transitive law for logical equivalence.


Theorem	bitr2i 191	An inference from transitive law for logical equivalence.


Theorem	bitr3i 192	An inference from transitive law for logical equivalence.


Theorem	bitr4i 193	An inference from transitive law for logical equivalence.


Theorem	3bitri 194	A chained inference from transitive law for logical equivalence.


Theorem	3bitrri 195	A chained inference from transitive law for logical equivalence.


Theorem	3bitr2i 196	A chained inference from transitive law for logical equivalence.


Theorem	3bitr2ri 197	A chained inference from transitive law for logical equivalence.


Theorem	3bitr3i 198	A chained inference from transitive law for logical equivalence.


Theorem	3bitr3ri 199	A chained inference from transitive law for logical equivalence.


Theorem	3bitr4i 200	A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence.

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