mmtheorems2 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 101-200 - Page 2 of 123

Type	Label	Description
Statement
Theorem	pm2.52 101	Theorem *2.52 of [WhiteheadRussell] p. 107.
|- (-. (ph -> ps) -> (-. ph -> -. ps))
Theorem	pm2.521 102	Theorem *2.521 of [WhiteheadRussell] p. 107.
|- (-. (ph -> ps) -> (ps -> ph))
Theorem	pm2.24i 103	Inference version of pm2.24 79.
|- ph   =>   |- (-. ph -> ps)
Theorem	pm2.24d 104	Deduction version of pm2.21 76.
|- (ph -> ps)   =>   |- (ph -> (-. ps -> ch))
Theorem	mto 105	The rule of modus tollens.
|- -. ps   &   |- (ph -> ps)   =>   |- -. ph
Theorem	mtoi 106	Modus tollens inference.
|- -. ch   &   |- (ph -> (ps -> ch))   =>   |- (ph -> -. ps)
Theorem	mtod 107	Modus tollens deduction.
|- (ph -> -. ch)   &   |- (ph -> (ps -> ch))   =>   |- (ph -> -. ps)
Theorem	mt2 108	A rule similar to modus tollens.
|- ps   &   |- (ph -> -. ps)   =>   |- -. ph
Theorem	mt2i 109	Modus tollens inference.
|- ch   &   |- (ph -> (ps -> -. ch))   =>   |- (ph -> -. ps)
Theorem	mt2d 110	Modus tollens deduction.
|- (ph -> ch)   &   |- (ph -> (ps -> -. ch))   =>   |- (ph -> -. ps)
Theorem	mt3 111	A rule similar to modus tollens.
|- -. ps   &   |- (-. ph -> ps)   =>   |- ph
Theorem	mt3i 112	Modus tollens inference.
|- -. ch   &   |- (ph -> (-. ps -> ch))   =>   |- (ph -> ps)
Theorem	mt3d 113	Modus tollens deduction.
|- (ph -> -. ch)   &   |- (ph -> (-. ps -> ch))   =>   |- (ph -> ps)
Theorem	mt4d 114	Modus tollens deduction.
|- (ph -> ps)   &   |- (ph -> (-. ch -> -. ps))   =>   |- (ph -> ch)
Theorem	nsyl 115	A negated syllogism inference.
|- (ph -> -. ps)   &   |- (ch -> ps)   =>   |- (ph -> -. ch)
Theorem	nsyld 116	A negated syllogism deduction.
|- (ph -> (ps -> -. ch))   &   |- (ph -> (ta -> ch))   =>   |- (ph -> (ps -> -. ta))
Theorem	nsyl2 117	A negated syllogism inference.
|- (ph -> -. ps)   &   |- (-. ch -> ps)   =>   |- (ph -> ch)
Theorem	nsyl3 118	A negated syllogism inference.
|- (ph -> -. ps)   &   |- (ch -> ps)   =>   |- (ch -> -. ph)
Theorem	nsyl4 119	A negated syllogism inference.
|- (ph -> ps)   &   |- (-. ph -> ch)   =>   |- (-. ch -> ps)
Theorem	nsyli 120	A negated syllogism inference.
|- (ph -> (ps -> ch))   &   |- (th -> -. ch)   =>   |- (ph -> (th -> -. ps))
Theorem	pm3.2im 121	Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives. (The proof was shortened by Josh Purinton, 29-Dec-2000.)
|- (ph -> (ps -> -. (ph -> -. ps)))
Theorem	mth8 122	Theorem 8 of [Margaris] p. 60. (The proof was shortened by Josh Purinton, 29-Dec-2000.)
|- (ph -> (-. ps -> -. (ph -> ps)))
Theorem	pm2.61 123	Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an antecedent.
|- ((ph -> ps) -> ((-. ph -> ps) -> ps))
Theorem	pm2.61i 124	Inference eliminating an antecedent.
|- (ph -> ps)   &   |- (-. ph -> ps)   =>   |- ps
Theorem	pm2.61d 125	Deduction eliminating an antecedent.
|- (ph -> (ps -> ch))   &   |- (ph -> (-. ps -> ch))   =>   |- (ph -> ch)
Theorem	pm2.61d1 126	Inference eliminating an antecedent.
|- (ph -> (ps -> ch))   &   |- (-. ps -> ch)   =>   |- (ph -> ch)
Theorem	pm2.61d2 127	Inference eliminating an antecedent.
|- (ph -> (-. ps -> ch))   &   |- (ps -> ch)   =>   |- (ph -> ch)
Theorem	pm2.61ii 128	Inference eliminating two antecedents. (The proof was shortened by Josh Purinton, 29-Dec-2000.)
|- (-. ph -> (-. ps -> ch))   &   |- (ph -> ch)   &   |- (ps -> ch)   =>   |- ch
Theorem	pm2.61nii 129	Inference eliminating two antecedents.
|- (ph -> (ps -> ch))   &   |- (-. ph -> ch)   &   |- (-. ps -> ch)   =>   |- ch
Theorem	pm2.61iii 130	Inference eliminating three antecedents.
|- (-. ph -> (-. ps -> (-. ch -> th)))   &   |- (ph -> th)   &   |- (ps -> th)   &   |- (ch -> th)   =>   |- th
Theorem	pm2.6 131	Theorem *2.6 of [WhiteheadRussell] p. 107.
|- ((-. ph -> ps) -> ((ph -> ps) -> ps))
Theorem	pm2.65 132	Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction.
|- ((ph -> ps) -> ((ph -> -. ps) -> -. ph))
Theorem	pm2.65i 133	Inference rule for proof by contradiction.
|- (ph -> ps)   &   |- (ph -> -. ps)   =>   |- -. ph
Theorem	pm2.65d 134	Deduction rule for proof by contradiction.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> -. ch))   =>   |- (ph -> -. ps)
Theorem	ja 135	Inference joining the antecedents of two premises. (The proof was shortened by O'Cat, 19-Feb-2008.)
|- (-. ph -> ch)   &   |- (ps -> ch)   =>   |- ((ph -> ps) -> ch)
Theorem	jc 136	Inference joining the consequents of two premises.
|- (ph -> ps)   &   |- (ph -> ch)   =>   |- (ph -> -. (ps -> -. ch))
Theorem	pm3.26im 137	Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112.
|- (-. (ph -> -. ps) -> ph)
Theorem	pm3.27im 138	Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112.
|- (-. (ph -> -. ps) -> ps)
Theorem	impt 139	Importation theorem expressed with primitive connectives.
|- ((ph -> (ps -> ch)) -> (-. (ph -> -. ps) -> ch))
Theorem	expt 140	Exportation theorem expressed with primitive connectives.
|- ((-. (ph -> -. ps) -> ch) -> (ph -> (ps -> ch)))
Theorem	impi 141	An importation inference.
|- (ph -> (ps -> ch))   =>   |- (-. (ph -> -. ps) -> ch)
Theorem	expi 142	An exportation inference. (The proof was shortened by O'Cat, 28-Nov-2008.)
|- (-. (ph -> -. ps) -> ch)   =>   |- (ph -> (ps -> ch))
Theorem	bijust 143	Theorem used to justify definition of biconditional df-bi 145. (The proof was shortened by Josh Purinton, 29-Dec-2000.)
|- -. ((ph -> ph) -> -. (ph -> ph))
Logical equivalence
Syntax	wb 144	Extend our wff definition to include the biconditional connective.
wff (ph <-> ps)
Definition	df-bi 145	This is our first definition, which introduces and defines the biconditional connective <->. We define a wff of the form (ph <-> ps) as an abbreviation for -. ((ph -> ps) -> -. (ps -> ph)). 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 222 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 783) 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 143. It is impossible to use df-bi 145 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 143 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 156, dfbi2 517, and dfbi3 673 for theorems suggesting typical textbook definitions of <->, showing that our definition has the properties we expect. Theorem dfbi 518 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.
|- -. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
Theorem	bi1 146	Property of the biconditional connective.
|- ((ph <-> ps) -> (ph -> ps))
Theorem	bi2 147	Property of the biconditional connective.
|- ((ph <-> ps) -> (ps -> ph))
Theorem	bi3 148	Property of the biconditional connective.
|- ((ph -> ps) -> ((ps -> ph) -> (ph <-> ps)))
Theorem	biimpi 149	Infer an implication from a logical equivalence.
|- (ph <-> ps)   =>   |- (ph -> ps)
Theorem	biimpri 150	Infer a converse implication from a logical equivalence.
|- (ph <-> ps)   =>   |- (ps -> ph)
Theorem	biimpd 151	Deduce an implication from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (ps -> ch))
Theorem	biimprd 152	Deduce a converse implication from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (ch -> ps))
Theorem	biimpcd 153	Deduce a commuted implication from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ps -> (ph -> ch))
Theorem	biimprcd 154	Deduce a converse commuted implication from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ch -> (ph -> ps))
Theorem	impbii 155	Infer an equivalence from an implication and its converse.
|- (ph -> ps)   &   |- (ps -> ph)   =>   |- (ph <-> ps)
Theorem	dfbi1 156	Relate the biconditional connective to primitive connectives. See dfbi1gb 157 for an unusual version proved directly from axioms.
|- ((ph <-> ps) <-> -. ((ph -> ps) -> -. (ps -> ph)))
Theorem	dfbi1gb 157	This proof of dfbi1 156, 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 145, compared to over 800 steps were the proof of dfbi1 156 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.
|- ((ph <-> ps) <-> -. ((ph -> ps) -> -. (ps -> ph)))
Theorem	bi2.04 158	Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122.
|- ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch)))
Theorem	notnot 159	Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117.
|- (ph <-> -. -. ph)
Theorem	pm4.8 160	Theorem *4.8 of [WhiteheadRussell] p. 122.
|- ((ph -> -. ph) <-> -. ph)
Theorem	pm4.81 161	Theorem *4.81 of [WhiteheadRussell] p. 122.
|- ((-. ph -> ph) <-> ph)
Theorem	con1b 162	Contraposition. Bidirectional version of con1 92.
|- ((-. ph -> ps) <-> (-. ps -> ph))
Theorem	con2b 163	Contraposition. Bidirectional version of con2 90.
|- ((ph -> -. ps) <-> (ps -> -. ph))
Theorem	con34b 164	Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116.
|- ((ph -> ps) <-> (-. ps -> -. ph))
Theorem	pm5.4 165	Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125.
|- ((ph -> (ph -> ps)) <-> (ph -> ps))
Theorem	imdi 166	Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125.
|- ((ph -> (ps -> ch)) <-> ((ph -> ps) -> (ph -> ch)))
Theorem	pm5.41 167	Theorem *5.41 of [WhiteheadRussell] p. 125.
|- (((ph -> ps) -> (ph -> ch)) <-> (ph -> (ps -> ch)))
Theorem	biid 168	Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117.
|- (ph <-> ph)
Theorem	biidd 169	Principle of identity with antecedent.
|- (ph -> (ps <-> ps))
Theorem	bicomi 170	Inference from commutative law for logical equivalence.
|- (ph <-> ps)   =>   |- (ps <-> ph)
Theorem	bitri 171	An inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ps <-> ch)   =>   |- (ph <-> ch)
Theorem	bitr2i 172	An inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ps <-> ch)   =>   |- (ch <-> ph)
Theorem	bitr3i 173	An inference from transitive law for logical equivalence.
|- (ps <-> ph)   &   |- (ps <-> ch)   =>   |- (ph <-> ch)
Theorem	bitr4i 174	An inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ch <-> ps)   =>   |- (ph <-> ch)
Theorem	3bitri 175	A chained inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ps <-> ch)   &   |- (ch <-> th)   =>   |- (ph <-> th)
Theorem	3bitrri 176	A chained inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ps <-> ch)   &   |- (ch <-> th)   =>   |- (th <-> ph)
Theorem	3bitr2i 177	A chained inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ch <-> ps)   &   |- (ch <-> th)   =>   |- (ph <-> th)
Theorem	3bitr2ri 178	A chained inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ch <-> ps)   &   |- (ch <-> th)   =>   |- (th <-> ph)
Theorem	3bitr3i 179	A chained inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ph <-> ch)   &   |- (ps <-> th)   =>   |- (ch <-> th)
Theorem	3bitr3ri 180	A chained inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ph <-> ch)   &   |- (ps <-> th)   =>   |- (th <-> ch)
Theorem	3bitr4i 181	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.
|- (ph <-> ps)   &   |- (ch <-> ph)   &   |- (th <-> ps)   =>   |- (ch <-> th)
Theorem	3bitr4ri 182	A chained inference from transitive law for logical equivalence.
|- (ph <-> ps)   &   |- (ch <-> ph)   &   |- (th <-> ps)   =>   |- (th <-> ch)
Theorem	imbi2i 183	Introduce an antecedent to both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ((ch -> ph) <-> (ch -> ps))
Theorem	imbi1i 184	Introduce a consequent to both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ((ph -> ch) <-> (ps -> ch))
Theorem	notbii 185	Negate both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- (-. ph <-> -. ps)
Theorem	imbi12i 186	Join two logical equivalences to form equivalence of implications.
|- (ph <-> ps)   &   |- (ch <-> th)   =>   |- ((ph -> ch) <-> (ps -> th))
Theorem	mpbi 187	An inference from a biconditional, related to modus ponens.
|- ph   &   |- (ph <-> ps)   =>   |- ps
Theorem	mpbir 188	An inference from a biconditional, related to modus ponens.
|- ps   &   |- (ph <-> ps)   =>   |- ph
Theorem	mtbi 189	An inference from a biconditional, related to modus tollens.
|- -. ph   &   |- (ph <-> ps)   =>   |- -. ps
Theorem	mtbir 190	An inference from a biconditional, related to modus tollens.
|- -. ps   &   |- (ph <-> ps)   =>   |- -. ph
Theorem	mpbii 191	An inference from a nested biconditional, related to modus ponens.
|- ps   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> ch)
Theorem	mpbiri 192	An inference from a nested biconditional, related to modus ponens.
|- ch   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> ps)
Theorem	mpbid 193	A deduction from a biconditional, related to modus ponens.
|- (ph -> ps)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> ch)
Theorem	mpbird 194	A deduction from a biconditional, related to modus ponens.
|- (ph -> ch)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> ps)
Theorem	a1bi 195	Inference rule introducing a theorem as an antecedent.
|- ph   =>   |- (ps <-> (ph -> ps))
Theorem	sylib 196	A mixed syllogism inference from an implication and a biconditional.
|- (ph -> ps)   &   |- (ps <-> ch)   =>   |- (ph -> ch)
Theorem	sylbi 197	A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition.
|- (ph <-> ps)   &   |- (ps -> ch)   =>   |- (ph -> ch)
Theorem	sylibr 198	A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition.
|- (ph -> ps)   &   |- (ch <-> ps)   =>   |- (ph -> ch)
Theorem	sylbir 199	A mixed syllogism inference from a biconditional and an implication.
|- (ps <-> ph)   &   |- (ps -> ch)   =>   |- (ph -> ch)
Theorem	sylibd 200	A syllogism deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (ps -> th))